parent	child	child_depth	certainty
math	math.la	0	0.99
Linear algebra
math.la	math.la.c.linsys	1	0.99
Linear systems of equations
math.la.c.linsys	math.la.c.linsys.terminology	2	0.99
Basic terminology
math.la.c.linsys.terminology	math.la.d.lineqn	3	0.99
Definition of linear equation
math.la.c.linsys.terminology	math.la.d.lineqn.coeff	3	0.98
Definition of coefficients of a linear equation
math.la.c.linsys.terminology	math.la.d.lineqn.soln	3	0.97
Definition of solution to a linear equation
math.la.c.linsys.terminology	math.la.d.linsys	3	0.96
Definition of system of linear equations
math.la.c.linsys.terminology	math.la.d.linsys.soln	3	0.95
Definition of solution to a system of linear equations
math.la.c.linsys.terminology	math.la.d.linsys.soln_set	3	0.94
Definition of solution set of a system of linear equations
math.la.c.linsys	math.la.c.linsys.soln_set.parameter	2	0.98
Parametric form of the solution set of a system of linear equations
math.la.c.linsys	math.la.c.linsys.soln_set.vec	2	0.97
Parametric vector form of the solution set of a system of linear equations
math.la.c.linsys	math.la.c.linsys.gauss	2	0.96
Gaussian elimination as a method to solve a linear system
math.la.c.linsys	math.la.d.linsys.echelon	2	0.95
Definition of echelon form of a linear system
math.la.c.linsys	math.la.t.linsys.echelon.free	2	0.94
All echelon forms of a linear system have the same free variables
math.la.c.linsys	math.la.d.linsys.op	2	0.93
Definition of equation operations on a linear system
math.la.c.linsys	math.la.t.linsys.op	2	0.92
Equation operations on a linear system give an equivalent system.
math.la.c.linsys	math.la.d.linsys.equiv	2	0.91
Definition of equivalent systems of linear equations
math.la.c.linsys	math.la.c.linsys.geometric	2	0.90
The geometry of linear systems
math.la.c.linsys.geometric	math.la.c.linsys.2x2.geometric	3	0.99
Geometric picture of a 2-by-2 linear system
math.la.c.linsys.geometric	math.la.c.lineqn.3.geometric	3	0.98
Geometric picture of the solution set of a linear equation in 3 unknowns
math.la.c.linsys.geometric	math.la.c.linsys.3x3.geometric	3	0.97
Geometric picture of a 3-by-3 linear system
math.la.c.linsys	math.la.d.linsys.consistent	2	0.89
Definition of consistent linear system
math.la.c.linsys	math.la.d.linsys.inconsistent	2	0.88
Definition of inconsistent linear system
math.la.c.linsys	math.la.d.linsys.homog	2	0.87
Definition of homogeneous linear system of equations
math.la.c.linsys	math.la.d.linsys.homog.consistent	2	0.86
Homogeneous linear systems are consistent.
math.la.c.linsys	math.la.d.linsys.homog.nontrivial	2	0.85
Definition of nontrivial solution to a homogeneous linear system of equations
math.la.c.linsys	math.la.d.linsys.homog.trivial	2	0.84
Definition of trivial solution to a homogeneous linear system of equations
math.la.c.linsys	math.la.t.linsys.homog.nontrivial	2	0.83
A homogeneous system has a nontrivial solution if and only if it has a free variable.
math.la.c.linsys	math.la.c.linsys.soln.number	2	0.82
The number of solutions to a linear system
math.la.c.linsys.soln.number	math.la.t.linsys.zoi	3	0.99
Linear systems have 0, 1, or infinitely many solutions.
math.la.c.linsys.soln.number	math.la.t.linsys.consistent.i	3	0.98
A consistent system with more variables than equations has infinitely many solutions.
math.la.c.linsys.soln.number	math.la.t.linsys.homog.i	3	0.97
A homogeneous system with more variables than equations has infinitely many solutions.
math.la.c.linsys	math.la.d.linsys.variable.dependent	2	0.81
Definition of basic/dependent/leading variable in a linear system
math.la.c.linsys	math.la.d.linsys.variable.independent	2	0.80
Definition of free/independent variable in a linear system
math.la.c.linsys	math.la.t.linsys.soln.vector	2	0.79
Theorem describing the vector form of sulutions to a linear system.
math.la.c.linsys	math.la.t.linsys.nonhomog.particular_plus_homog	2	0.78
The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.
math.la.c.linsys	math.la.d.linsys.ill_conditioned	2	0.77
Definition of ill-conditioned linear system
math.la	math.la.c.mat	1	0.98
Matrices
math.la.c.mat	math.la.c.mat.basics	2	0.99
Basic terminology and notation
math.la.c.mat.basics	math.la.d.mat	3	0.99
Definition of matrix
math.la.c.mat.basics	math.la.d.mat.entry	3	0.98
Notation for entry of matrix
math.la.c.mat.basics	math.la.d.mat.size	3	0.97
Definition of size of a matrix
math.la.c.mat.basics	math.la.d.mat.m_by_n	3	0.96
Definition of m by n matrix
math.la.c.mat.basics	math.la.d.mat.m_by_n.set	3	0.95
Notation for the set of m by n matrices
math.la.c.mat.basics	math.la.d.mat.square	3	0.94
Definition of square matrix
math.la.c.mat.basics	math.la.d.mat.thediagonal	3	0.93
Definition of the (main) diagonal of a matrix
math.la.c.mat.basics	math.la.d.mat.diagonal	3	0.92
Definition of diagonal matrix
math.la.c.mat.basics	math.la.d.mat.identity	3	0.91
Definition of identity matrix
math.la.c.mat.basics	math.la.d.mat.z	3	0.90
Definition of 0 matrix
math.la.c.mat.basics	math.la.d.mat.equal	3	0.89
Definition of equality of matrices
math.la.c.mat	math.la.c.mat.operations	2	0.98
Operations on matrices
math.la.c.mat.operations	math.la.c.mat.sum	3	0.99
Addition
math.la.c.mat.sum	math.la.d.mat.sum	4	0.99
Definition of sum of matrices
math.la.c.mat.sum	math.la.t.mat.add.commut_assoc	4	0.98
Matrix addition is commutative and associative.
math.la.c.mat.operations	math.la.c.mat.conjugate	3	0.98
Conjugation
math.la.c.mat.conjugate	math.la.d.mat.conjugate	4	0.99
Definition of conjugate of a matrix
math.la.c.mat.conjugate	math.la.t.mat.conjugate.involution	4	0.98
Matrix conjugation is an involution.
math.la.c.mat.conjugate	math.la.t.mat.sum.conjugate	4	0.97
The conjugate of the sum of matrices is the sum of the conjugates.
math.la.c.mat.conjugate	math.la.t.mat.scalar.conjugate	4	0.96
The conjugate of a matrix-scalar product is the product of the conjugates.
math.la.c.mat.conjugate	math.la.t.mat.mult.conjugate	4	0.95
The conjugate of a product of matrices is the product of the conjugates.
math.la.c.mat.operations	math.lac.mat.scalar.mult	3	0.97
Scalar multiplication
math.lac.mat.scalar.mult	math.la.d.mat.scalar.mult	4	0.99
Definition of matrix-scalar multiplication
math.lac.mat.scalar.mult	math.la.t.mat.scalar.mult.commut_assoc	4	0.98
Matrix-scalar multiplication is commutative, associative, and distributive.
math.lac.mat.scalar.mult	math.la.t.mat.scalar.prod.commut	4	0.97
Matrix-scalar product is commutative
math.la.c.mat.operations	math.la.c.mat.row_op	3	0.96
Row operations
math.la.c.mat.row_op	math.la.d.mat.row_op	4	0.99
Definition of row operations on a matrix
math.la.c.mat.row_op	math.la.d.mat.pivot	4	0.98
Definition of pivot
math.la.c.mat.row_op	math.la.d.mat.pivot_col	4	0.97
Definition of pivot column
math.la.c.mat.row_op	math.la.d.mat.pivot_position	4	0.96
Definition of pivot position
math.la.c.mat.row_op	math.la.d.mat.row_equiv	4	0.95
Definition of row equivalent matrices
math.la.c.mat.row_op	math.la.t.mat.row_equiv.equiv	4	0.94
Row equivalence is an equivalence relation
math.la.c.mat.row_op	math.la.d.mat.row.leading	4	0.93
Definition of leading entry in a row of a matrix
math.la.c.mat.operations	math.la.c.mat.vec.prod	3	0.95
Matrix-vector products
math.la.c.mat.vec.prod	math.la.d.mat.vec.prod	4	0.99
Definition of matrix-vector product, as a linear combination of column vectors
math.la.c.mat.vec.prod	math.la.t.mat.vec.prod.unique	4	0.98
If two matrices have equal products with all vectors, then the matrices are equal.
math.la.c.mat.vec.prod	math.la.e.mat.vec.prod	4	0.97
Example of matrix-vector product, as a linear combination of column vectors
math.la.c.mat.vec.prod	math.la.d.mat.vec.prod.coord	4	0.96
Definition of matrix-vector product, each entry separately
math.la.c.mat.vec.prod	math.la.e.mat.vec.prod.coord	4	0.95
Example of matrix-vector product, each entry separately
math.la.c.mat.vec.prod	math.la.t.mat.vec.prod.assoc	4	0.94
Matrix-vector product is associative
math.la.c.mat.operations	math.la.c.mat.mult	3	0.94
Multiplication
math.la.c.mat.mult	math.la.d.mat.mult.col	4	0.99
Definition of matrix multiplication in terms of column vectors
math.la.c.mat.mult	math.la.d.mat.mult.coord	4	0.98
Definition of matrix multiplication, each entry separately
math.la.c.mat.mult	math.la.t.mat.mult.coord	4	0.97
Theorem describing matrix multiplication, each entry separately
math.la.c.mat.mult	math.la.t.mat.mult.z	4	0.96
Any matrix times the 0 matrix equals the 0 matrix.
math.la.c.mat.mult	math.la.e.mat.mult.2x2	4	0.95
Example of multiplying 2x2 matrices
math.la.c.mat.mult	math.la.e.mat.mult.3x3	4	0.94
Example of multiplying 3x3 matrices
math.la.c.mat.mult	math.la.e.mat.mult.nonsquare	4	0.93
Example of multiplying nonsquare matrices
math.la.c.mat.mult	math.la.e.mat.mult	4	0.92
Example of multiplying matrices
math.la.c.mat.mult	math.la.t.mat.mult.assoc	4	0.91
Matrix multiplication is associative.
math.la.c.mat.mult	math.la.t.mat.mult.distributive	4	0.90
Matrix multiplication is distributive over matrix addition.
math.la.c.mat.mult	math.la.t.mat.mult.identity	4	0.89
The identity matrix is the identity for matrix multiplication.
math.la.c.mat.mult	math.la.c.mat.mult.commut	4	0.88
Matrix multiplication is not commutative in general.
math.la.c.mat.mult	math.la.c.mat.mult.cancellation	4	0.87
For matrices, AB=AC does not imply B=C in general.
math.la.c.mat.mult	math.la.c.mat.mult.zero_divisor	4	0.86
For matrices, AB=0 does not imply A=0 or B=0 in general.
math.la.c.mat.mult	math.la.t.mat.mult.row.col	4	0.85
Matrix multiplication can be viewed as the dot product of a row vector of column vectors with a column vector of row vectors
math.la.c.mat.operations	math.la.c.mat.transpose	3	0.93
Transpose and adjoint
math.la.c.mat.transpose	math.la.d.mat.transpose	4	0.99
Definition of transpose of a matrix
math.la.c.mat.transpose	math.la.t.mat.transpose.involution	4	0.98
Matrix transpose is an involution.
math.la.c.mat.transpose	math.la.t.mat.transpose.conjugate	4	0.97
The conjugate of the transpose is the transpose of the conjugate.
math.la.c.mat.transpose	math.la.d.mat.adjoint	4	0.96
Definition of adjoint (conjugate transpose)
math.la.c.mat.transpose	math.la.t.mat.sum.adjoint	4	0.95
The adjoint of a sum is the sum of the adjoints.
math.la.c.mat.transpose	math.la.t.mat.scalar.adjoint	4	0.94
The adjoint of a matrix-scalar product is the product of the adjoint and the conjugate.
math.la.c.mat.transpose	math.la.t.mat.adjoint.involution	4	0.93
Matrix adjoint is an involution.
math.la.c.mat.transpose	math.la.t.mat.sum.transpose	4	0.92
The transpose of a sum of matrices is the sum of the transposes.
math.la.c.mat.transpose	math.la.t.mat.scalar.transpose	4	0.91
Transpose commutes with scalar multiplication.
math.la.c.mat.transpose	math.la.t.mat.mult.transpose	4	0.90
The transpose of a product of matrices is the product of the transposes in reverse order.
math.la.c.mat.transpose	math.la.t.mat.mult.adjoint	4	0.89
The adjoint of a product of matrices is the product of the adjoints in reverse order.
math.la.c.mat.operations	math.la.c.mat.inv	3	0.92
Inverse
math.la.c.mat.inv	math.la.d.mat.inv.left	4	0.99
Definition of left inverse of a matrix
math.la.c.mat.inv	math.la.d.mat.inv.right	4	0.98
Definition of right inverse of a matrix
math.la.c.mat.inv	math.la.d.mat.inv	4	0.97
Definition of inverse of a matrix
math.la.c.mat.inv	math.la.d.mat.invertible	4	0.96
Definition of invertible matrix
math.la.c.mat.inv	math.la.t.mat.inv.leftright	4	0.95
If a matrix has both a left and a right inverse, then the two are equal.
math.la.c.mat.inv	math.la.t.mat.inv.oneside	4	0.94
If a square matrix has a one-sided inverse, then it is a two-sided inverse.
math.la.c.mat.inv	math.la.t.mat.inv.2x2	4	0.93
Formula for the inverse of a 2-by-2 matrix.
math.la.c.mat.inv	math.la.t.mat.inv.involution	4	0.92
Matrix inverse is an involution.
math.la.c.mat.inv	math.la.t.mat.inv.shoesandsocks	4	0.91
For n-by-n invertible matrices A and B, the product AB is invertible, and (AB)^-1=B^-1 A^-1.
math.la.c.mat.inv	math.la.t.mat.prod.nonsingular	4	0.90
The product of square matrices is nonsingular if and only if each factor is nonsingular.
math.la.c.mat.inv	math.la.t.mat.inv.scalar	4	0.89
The inverse of a scalar multiple is the reciprocal times the inverse.
math.la.c.mat.inv	math.la.t.mat.inv.transpose	4	0.88
Matrix transpose commutes with matrix inverse.
math.la.c.mat.inv	math.la.t.mat.inv.augmented	4	0.87
The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.
math.la.c.mat.inv	math.la.e.mat.inv.2x2.row_reduce	4	0.86
Example of finding the inverse of a 2-by-2 matrix by row reducing the augmented matrix
math.la.c.mat.inv	math.la.e.mat.inv.2x2.formula	4	0.85
Example of finding the inverse of a 2-by-2 matrix by using a formula
math.la.c.mat.inv	math.la.e.mat.inv.3x3.row_reduce	4	0.84
Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix
math.la.c.mat.inv	math.la.e.mat.inv.3x3.cramer	4	0.83
Example of finding the inverse of a 3-by-3 matrix by using Cramer's rule
math.la.c.mat.inv	math.la.t.eqn.mat.inv	4	0.82
The inverse of a matrix can be used to solve a linear system.
math.la.c.mat.inv	math.la.t.mat.inv.unique	4	0.81
Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1.
math.la.c.mat.inv	math.la.d.mat.inv.generalized	4	0.80
Definition of generalized inverse of a matrix
math.la.c.mat	math.la.c.mat.types	2	0.97
Particular types of matrices
math.la.c.mat.types	math.la.c.mat.echelon	3	0.99
Echelon matrices
math.la.c.mat.echelon	math.la.e.mat.echelon.of	4	0.99
Example of putting a matrix in echelon form
math.la.c.mat.echelon	math.la.e.mat.echelon.of.pivot	4	0.98
Example of putting a matrix in echelon form and identifying the pivot columns
math.la.c.mat.echelon	math.la.d.mat.echelon	4	0.97
Definition of (echelon matrix/matrix in (row) echelon form)
math.la.c.mat.echelon	math.la.c.mat.gaussjordan	4	0.96
Gauss-Jordan procedure to put a matrix into reduced row echelon form
math.la.c.mat.echelon	math.la.e.mat.echelon	4	0.95
Example of (echelon matrix/matrix in (row) echelon form)
math.la.c.mat.echelon	math.la.d.mat.echelon.of	4	0.94
Definition of (row) echelon form of a matrix
math.la.c.mat.echelon	math.la.d.mat.rref	4	0.93
Definition of matrix in reduced row echelon form
math.la.c.mat.echelon	math.la.d.mat.rref.of	4	0.92
Definition of reduced row echelon form of a matrix
math.la.c.mat.echelon	math.la.d.mat.row_reduce	4	0.91
Definition of row reduce a matrix
math.la.c.mat.echelon	math.la.e.mat.row_reduce.3x3	4	0.90
Example of row reducing a 3-by-3 matrix
math.la.c.mat.echelon	math.la.e.mat.row_reduce.4x4	4	0.89
Example of row reducing a 4-by-4 matrix
math.la.c.mat.echelon	math.la.t.mat.rref.exists	4	0.88
Every matrix is row-equivalent to a matrix in reduced row echelon form.
math.la.c.mat.echelon	math.la.t.mat.rref.unique	4	0.87
Every matrix is row-equivalent to only one matrix in reduced row echelon form.
math.la.c.mat.echelon	math.la.d.mat.erref.of	4	0.86
Definition of extended reduced row echelon form of a matrix
math.la.c.mat.echelon	math.la.t.mat.erref.of	4	0.85
Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix
math.la.c.mat.echelon	math.la.t.mat.erref.spaces	4	0.84
Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix
math.la.c.mat.echelon	math.la.t.mat.erref.dimension	4	0.83
Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix
math.la.c.mat.types	math.la.d.mat.unit	3	0.98
Definition of unit matrix
math.la.c.mat.types	math.la.d.mat.permutation	3	0.97
Definition of permutation matrix
math.la.c.mat.types	math.la.c.mat.elementary	3	0.96
Elementary matrices
math.la.c.mat.elementary	math.la.d.mat.elementary	4	0.99
Definition of elementary matrix
math.la.c.mat.elementary	math.la.t.mat.elementary.prod	4	0.98
A nonsingular matrix can be written as a product of elementary matrices.
math.la.c.mat.elementary	math.la.t.mat.mult.elementary	4	0.97
Row operations are given by multiplication by elementary matrices.
math.la.c.mat.elementary	math.la.d.mat.elementary.inv	4	0.96
Elementary matrices are invertible/nonsingular.
math.la.c.mat.types	math.la.c.mat.triamgular	3	0.95
Triangular matrices
math.la.c.mat.triamgular	math.la.d.mat.triangular.upper	4	0.99
Definition of an upper triangular matrix
math.la.c.mat.triamgular	math.la.d.mat.triangular.lower	4	0.98
Definition of a lower triangular matrix
math.la.c.mat.triamgular	math.la.t.mat.triangular.prod	4	0.97
The product of upper/lower triangular matrices is upper/lower triangular.
math.la.c.mat.triamgular	math.la.t.mat.triangular.inv	4	0.96
The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.
math.la.c.mat.triamgular	math.la.t.mat.triangular.unitary	4	0.95
Every square matrix is conjugate, via a unitary matrix, to an upper triangular matrix.
math.la.c.mat.types	math.la.c.mat.block	3	0.94
Block matrices
math.la.c.mat.block	math.la.d.mat.block	4	0.99
Definition of block/partitioned matrix
math.la.c.mat.block	math.la.c.mat.mult.block	4	0.98
Multiplication of block/partitioned matrices
math.la.c.mat.block	math.la.d.mat.block_diagonal	4	0.97
Definition of block diagonal matrix
math.la.c.mat.types	math.la.c.mat.symmetric	3	0.93
Symmetric matrices
math.la.c.mat.symmetric	math.la.d.mat.symmetric	4	0.99
Definition of symmetric matrix
math.la.c.mat.symmetric	math.la.t.mat.symmetric.square	4	0.98
Symmetric matrices are square.
math.la.c.mat.symmetric	math.la.t.mat.symmetric.eig.orthogonal	4	0.97
Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal.
math.la.c.mat.symmetric	math.la.t.mat.symmetric.spectral	4	0.96
The spectral theorem for symmetric matrices
math.la.c.mat.symmetric	math.la.t.mat.symmetric.spectraldecomposition	4	0.95
Formula for the spectral decomposition for a symmetric matrix
math.la.c.mat.symmetric	math.la.d.mat.diagonalizable.orthogonally	4	0.94
Definition of orthogonally diagonalizable matrix
math.la.c.mat.symmetric	math.la.t.mat.diagonalizable.orthogonally	4	0.93
A matrix is orthogonally diagonalizable if and only if it is symmetric.
math.la.c.mat.symmetric	math.la.d.mat.skewsymmetric	4	0.92
Definition of skew-symmetric matrix
math.la.c.mat.types	math.la.c.mat.nilpotent	3	0.92
Nilpotent matrices
math.la.c.mat.nilpotent	math.la.d.mat.nilpotent	4	0.99
Definition of nilpotent matrix
math.la.c.mat.nilpotent	math.la.t.mat.nilpotent.zo	4	0.98
Every nilpotent matrix is similar to one with 1 on subdiagonal blocks and all other entries 0.
math.la.c.mat.nilpotent	math.la.t.mat.nilpotent.eig	4	0.97
A matrix is nilpotent if and only if its only eigenvalue is 0.
math.la.c.mat.nilpotent	math.la.d.nilpotent.index	4	0.96
Definition of index of nilpotency
math.la.c.mat.nilpotent	math.la.t.mat.jordan.sum	4	0.95
Every square matrix is similar the sum of a diagonal and a nilpotent matrix.
math.la.c.mat.types	math.la.d.mat.orthogonal	3	0.91
Definition of orthogonal matrix
math.la.c.mat.types	math.la.c.mat.unitary	3	0.90
Unitary matrices
math.la.c.mat.unitary	math.la.d.mat.unitary	4	0.99
Definition of unitary matrix
math.la.c.mat.unitary	math.la.t.mat.unitary.inv	4	0.98
Unitary matrices are invertible.
math.la.c.mat.unitary	math.la.t.mat.unitary.col.orthogonal	4	0.97
Unitary matrices have orthogonal (orthonormal) rows/columns.
math.la.c.mat.unitary	math.la.t.mat.unitary.innerproduct	4	0.96
Unitary matrices preserve inner products.
math.la.c.mat.unitary	math.la.t.mat.unitary.basis.orthogonal	4	0.95
Unitary matrices preserve orthogonal (orthonormal) bases.
math.la.c.mat.types	math.la.d.mat.band	3	0.89
Definition of band matrix
math.la.c.mat.types	math.la.d.mat.vandermonde	3	0.88
Definition of Vandermonde matrix
math.la.c.mat.types	math.la.d.mat.markov	3	0.87
Definition of Markov matrix
math.la.c.mat.types	math.la.c.mat.hermitian	3	0.86
Hermitian matrices
math.la.c.mat.hermitian	math.la.d.mat.hermitian	4	0.99
Definition of Hermitian/self-adjoint matrix
math.la.c.mat.hermitian	math.la.d.mat.hermitian.innerproduct.cn	4	0.98
Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.
math.la.c.mat.types	math.la.c.mat.normal	3	0.85
Normal matrices
math.la.c.mat.normal	math.la.d.mat.normal	4	0.99
Definition of normal matrix
math.la.c.mat.normal	math.la.t.mat.normal.diagonalize	4	0.98
A matrix is orthogonally diagonalizable if and only if it is normal (The principal axis theorem).
math.la.c.mat.normal	math.la.t.mat.normal.eigenval	4	0.97
The eigenvectors of a normal matrix are an orthonormal basis.
math.la.c.mat	math.la.c.mat.equivalence	2	0.96
Matrix equivalence
math.la.c.mat.equivalence	math.la.d.mat.equiv	3	0.99
Definition of equivalent matrices
math.la.c.mat.equivalence	math.la.t.mat.equiv.map	3	0.98
Equivalent matrices represent the same linear transformation with resect to appropriate bases.
math.la.c.mat.equivalence	math.la.t.mat.equiv.diag	3	0.97
A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.
math.la.c.mat.equivalence	math.la.t.mat.equiv.rank	3	0.96
Two matrices of the same size are equivalent if and only if they have the same rank.
math.la.c.mat	math.la.c.mat.form	2	0.95
Canonical forms of matrices
math.la.c.mat.form	math.la.c.mat.diagonalize	3	0.99
Matrix diagonalization
math.la.c.mat.diagonalize	math.la.d.mat.diagonalization	4	0.99
Definition of matrix diagonalization
math.la.c.mat.diagonalize	math.la.d.mat.diagonalizable	4	0.98
Definition of diagonalizable matrix
math.la.c.mat.diagonalize	math.la.t.mat.diagonalizable	4	0.97
An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.
math.la.c.mat.diagonalize	math.la.t.mat.diagonalizable.eigenspace	4	0.96
An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n.
math.la.c.mat.diagonalize	math.la.t.mat.diagonalizable.charpoly	4	0.95
An n-by-n matrix is diagonalizable if and only if the characteristic polynomial factors completely, and the dimension of each eigenspace equals the (algebraic) multiplicity of the eigenvalue.
math.la.c.mat.diagonalize	math.la.t.mat.diagonalized_by	4	0.94
A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.
math.la.c.mat.diagonalize	math.la.t.mat.diagonalizable.basis	4	0.93
An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).
math.la.c.mat.diagonalize	math.la.t.mat.diagonalizable.distinct	4	0.92
An n-by-n matrix with n distinct eigenvalues is diagonalizable.
math.la.c.mat.diagonalize	math.la.t.mat.real.diagonalize.complex.2x2	4	0.91
Formula for diagonalizing a real 2-by-2 matrix with a complex eigenvalue.
math.la.c.mat.form	math.la.d.mat.hessenberg	3	0.98
Definition of Hessenberg form
math.la.c.mat.form	math.la.d.mat.jordan	3	0.97
Definition of Jordan form
math.la.c.mat.form	math.la.d.mat.rational	3	0.96
Definition of rational form
math.la.c.mat	math.la.c.mat.factor	2	0.94
Factorization of matrices
math.la.c.mat.factor	math.la.d.mat.svd	3	0.99
Definition of singular value decomposition (SVD)
math.la.c.mat.factor	math.la.c.mat.lu	3	0.98
LU decomposition
math.la.c.mat.lu	math.la.d.mat.lu	4	0.99
Definition of LU decomposition
math.la.c.mat.lu	math.la.t.mat.lu	4	0.98
Algorithm for computing an LU decomposition
math.la.c.mat.lu	math.la.d.mat.lu.reduced	4	0.97
Definition of reduced LU decomposition
math.la.c.mat.factor	math.la.d.mat.rank_factorization	3	0.97
Definition of rank factorization of a matrix
math.la.c.mat.factor	math.la.d.mat.cholesky	3	0.96
Definition of Cholesky decomposition
math.la.c.mat.factor	math.la.c.mat.qr	3	0.95
QR decomposition
math.la.c.mat.qr	math.la.d.mat.qr	4	0.99
Definition of QR decomposition
math.la.c.mat.qr	math.la.t.mat.qr	4	0.98
The QR decomposition of a nonsingular matrix exists.
math.la.c.mat.factor	math.la.d.mat.schur	3	0.94
Definition of Schur triangulation
math.la.c.mat	math.la.c.mat.similar	2	0.93
Similarity of matrices
math.la.c.mat.similar	math.la.d.mat.similar	3	0.99
Definition of similar matrices
math.la.c.mat.similar	math.la.t.mat.similar.equiv	3	0.98
Similarity of matrices in an equivalence relation.
math.la.c.mat.similar	math.la.d.mat.similar.transform	3	0.97
Definition of similarity transform
math.la.c.mat.similar	math.la.t.mat.similar.eig	3	0.96
Similar matrices have the same eigenvalues and the same characteristic polynomials.
math.la.c.mat.similar	math.la.t.mat.jordan	3	0.95
Every square matrix is similar to one in Jordan form.
math.la.c.mat	math.la.c.mat.nonsingular	2	0.92
Nonsingular matrices and equivalences
math.la.c.mat.nonsingular	math.la.d.mat.nonsingular.inv	3	0.99
Definition of nonsingular matrix: matrix is invertible
math.la.c.mat.nonsingular	math.la.d.mat.nonsingular.z	3	0.98
Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution
math.la.c.mat.nonsingular	math.la.d.mat.singular	3	0.97
Definition of singular matrix (not nonsingular)
math.la.c.mat.nonsingular	math.la.p.equiv.multiple	3	0.96
Proof of several equivalences for nonsingular matrix
math.la.c.mat.nonsingular	math.la.t.equiv.mat.eqn.unique	3	0.95
Equivalence theorem for nonsingular matrices: the equation Ax=b has a unique solution for all b.
math.la.c.mat.nonsingular	math.la.t.equiv.mat.eqn	3	0.94
Equivalence theorem for nonsingular matrices: the equation Ax=b has a solution for all b.
math.la.c.mat.nonsingular	math.la.t.equiv.mat.eqn.homog	3	0.93
Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.
math.la.c.mat.nonsingular	math.la.t.equiv.row.span	3	0.92
Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).
math.la.c.mat.nonsingular	math.la.t.equiv.col.span	3	0.91
Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).
math.la.c.mat.nonsingular	math.la.t.equiv.row.linindep	3	0.90
Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.
math.la.c.mat.nonsingular	math.la.t.equiv.col.linindep	3	0.89
Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.
math.la.c.mat.nonsingular	math.la.t.equiv.row.basis	3	0.88
Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).
math.la.c.mat.nonsingular	math.la.t.equiv.col.basis	3	0.87
Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).
math.la.c.mat.nonsingular	math.la.t.equiv.col.dim	3	0.86
Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.
math.la.c.mat.nonsingular	math.la.t.equiv.row.pivot	3	0.85
Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.
math.la.c.mat.nonsingular	math.la.t.equiv.identity	3	0.84
Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.
math.la.c.mat.nonsingular	math.la.t.equiv.inv	3	0.83
Equivalence theorem for nonsingular matrices: the matrix A has an inverse.
math.la.c.mat.nonsingular	math.la.t.equiv.inv.left	3	0.82
Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.
math.la.c.mat.nonsingular	math.la.t.equiv.inv.right	3	0.81
Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.
math.la.c.mat.nonsingular	math.la.t.equiv.transpose.inv	3	0.80
Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.
math.la.c.mat.nonsingular	math.la.t.equiv.lintrans.injective	3	0.79
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.
math.la.c.mat.nonsingular	math.la.t.equiv.lintrans.surjective	3	0.78
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.
math.la.c.mat.nonsingular	math.la.t.equiv.lintrans.inv	3	0.77
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.
math.la.c.mat.nonsingular	math.la.t.equiv.lintrans.isomorphism	3	0.76
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.
math.la.c.mat.nonsingular	math.la.t.equiv.det	3	0.75
Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.
math.la.c.mat.nonsingular	math.la.t.equiv.rank	3	0.74
Equivalence theorem for nonsingular matrices: the matrix A has rank n.
math.la.c.mat.nonsingular	math.la.t.equiv.nullspace	3	0.73
Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.
math.la.c.mat.nonsingular	math.la.t.equiv.nullity	3	0.72
Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.
math.la.c.mat.nonsingular	math.la.t.equiv.eig	3	0.71
Equivalence theorem for nonsingular matrices: the matrix A does not have 0 as an eigenvalue.
math.la.c.mat.nonsingular	math.la.t.equiv.change_of_basis	3	0.70
Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.
math.la.c.mat.nonsingular	math.la.t.equiv.identitymap	3	0.69
Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.
math.la.c.mat	math.la.c.mat.rank	2	0.91
Rank and mullity
math.la.c.mat.rank	math.la.d.mat.rank.column	3	0.99
Definition of column rank of a matrix
math.la.c.mat.rank	math.la.d.mat.rank	3	0.98
Definition of rank of a matrix
math.la.c.mat.rank	math.la.d.mat.nullity	3	0.97
Definition of nullity of a matrix
math.la.c.mat.rank	math.la.t.mat.rank.pivot	3	0.96
The rank of a matrix equals number of pivots in a reduced row echelon form.
math.la.c.mat.rank	math.la.t.mat.lintrans.rank	3	0.95
The rank of a matrix equals the rank of the linear transformation it represents.
math.la.c.mat.rank	math.la.t.mat.row_space.col_space	3	0.94
The row space and the column space of a matrix have the same dimension.
math.la.c.mat.rank	math.la.t.mat.ranknullity	3	0.93
If A is a matrix, then the rank of A plus the nullity of A equals the number of columns of A.
math.la.c.mat	math.la.c.mat.eig	2	0.90
Eigenvalues and eigenvectors
math.la.c.mat.eig	math.la.d.mat.eig	3	0.99
Definition of eigenvalue of a matrix
math.la.c.mat.eig	math.la.d.mat.eigvec	3	0.98
Definition of eigenvector of a matrix
math.la.c.mat.eig	math.la.c.mat.eigsp	3	0.97
Eigenspaces
math.la.c.mat.eigsp	math.la.d.mat.eigsp	4	0.99
Definition of eigenspace of a matrix
math.la.c.mat.eigsp	math.la.t.mat.eigsp.subsp	4	0.98
An eigenspace of a matrix is a nontrivial subspace.
math.la.c.mat.eigsp	math.la.t.mat.eigsp.nullspace	4	0.97
An eigenspace of a matrix is the null space of a related matrix.
math.la.c.mat.eig	math.la.t.mat.eig.exists	3	0.96
Every matrix has an eigenvalue over the complex numbers.
math.la.c.mat.eig	math.la.d.mat.eig.operations	3	0.95
Eigenvalues and operations on matrices
math.la.d.mat.eig.operations	math.la.t.mat.eig.scalar	4	0.99
The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.
math.la.d.mat.eig.operations	math.la.t.mat.eig.power	4	0.98
The eigenvalues of a power of a matrix are the power the eigenvalues.
math.la.d.mat.eig.operations	math.la.t.mat.eig.polynomial	4	0.97
The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.
math.la.d.mat.eig.operations	math.la.t.mat.eig.inv	4	0.96
The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.
math.la.d.mat.eig.operations	math.la.t.mat.eig.transpose	4	0.95
A matrix and its transpose have the same eigenvalues/characteristic polynomial.
math.la.c.mat.eig	math.la.t.mat.eigvec.linindep	3	0.94
Eigenvectors with distinct eigenvalues are linearly independent.
math.la.c.mat.eig	math.la.c.mat.eig.multiplicity	3	0.93
Multiplicity
math.la.c.mat.eig.multiplicity	math.la.d.mat.eig.multiplicity.algebraic	4	0.99
Definition of (algebraic) multiplicity of an eigenvalue
math.la.c.mat.eig.multiplicity	math.la.d.mat.eig.multiplicity.geometric	4	0.98
Definition of geometric multiplicity of an eigenvalue
math.la.c.mat.eig.multiplicity	math.la.d.mat.eig.number	4	0.97
An n-by-n matrix nas n (complex) eigenvalues, counted according to algebraic multiplicity.
math.la.c.mat.eig	math.la.c.mat.polynomial	3	0.92
Characteristic and minimal polynomials
math.la.c.mat.polynomial	math.la.d.mat.polynomial.apply	4	0.99
Definition of applying a polynomial to a square matrix
math.la.c.mat.polynomial	math.la.d.mat.charpoly	4	0.98
Definition of characteristic polynomial of a matrix
math.la.c.mat.polynomial	math.la.d.mat.charpoly.eqn	4	0.97
Definition of characteristic equation of a matrix
math.la.c.mat.polynomial	math.la.d.mat.minpoly	4	0.96
Definition of minimal polynomial of a matrix
math.la.c.mat.polynomial	math.la.d.mat.minpoly.exists	4	0.95
The minimal polynomial of a square matrix exists and is unique.
math.la.c.mat.polynomial	math.la.t.mat.charpoly.eig	4	0.94
The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation.
math.la.c.mat.polynomial	math.la.t.mat.charpoly.z	4	0.93
The characteristic polynomial applied to the matrix gives the 0 matrix.
math.la.c.mat.polynomial	math.la.d.mat.cayleyhamilton	4	0.92
The Cayley-Hamilton theorem for a matrix.
math.la.c.mat.eig	math.la.t.mat.eig.multiplicity.eigenspace	3	0.91
The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.
math.la.c.mat.eig	math.la.c.mat.eig.types	3	0.90
Particular types of matrices
math.la.c.mat.eig.types	math.la.t.mat.eig.triangular	4	0.99
The eigenvalues of a triangular matrix are the entries on the main diagonal.
math.la.c.mat.eig.types	math.la.t.mat.real.eig.cn	4	0.98
A matrix with real entries has eigenvalues occurring in conjugate pairs.
math.la.c.mat.eig.types	math.la.t.mat.hermitian.eig.real	4	0.97
Hermitian matrices have real eigenvalues.
math.la.c.mat.eig.types	math.la.t.mat.hermitian.eigvec.orthogonal	4	0.96
Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.
math.la.c.mat.eig.types	math.la.d.mat.positive_definite	4	0.95
Definition of positive-definite matrix
math.la.c.mat	math.la.c.mat.det	2	0.89
Determinants
math.la.c.mat.det	math.la.d.mat.trace	3	0.99
Definition of trace of a matrix
math.la.c.mat.det	math.la.c.mat.cofactor	3	0.98
Cofactors
math.la.c.mat.cofactor	math.la.d.mat.cofactor	4	0.99
Definition of cofactor/submatrix of a matrix
math.la.c.mat.cofactor	math.la.d.mat.det.cofactor	4	0.98
Definition of determinant of a matrix as a cofactor expansion across the first row
math.la.c.mat.cofactor	math.la.t.mat.det.cofactor.row	4	0.97
The determinant of a matrix can be computed as a cofactor expansion across any row.
math.la.c.mat.cofactor	math.la.t.mat.det.cofactor.col	4	0.96
The determinant of a matrix can be computed as a cofactor expansion down any column.
math.la.c.mat.cofactor	math.la.t.mat.inv.cofactors	4	0.95
The inverse of a matrix can be expressed in terms of its matrix of cofactors.
math.la.c.mat.cofactor	math.la.t.cramer	4	0.94
Cramer's rule
math.la.c.mat.det	math.la.c.mat.det.operations	3	0.97
Determinants and operations on matrices
math.la.c.mat.det.operations	math.la.d.mat.det.echelon	4	0.99
Definition of determinant of a matrix as a product of the diagonal entries in a non-scaled echelon form.
math.la.c.mat.det.operations	math.la.t.mat.det.echelon	4	0.98
The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.
math.la.c.mat.det.operations	math.la.d.mat.det.elementaryoperations	4	0.97
Definition of the determinant in terms of the effect of elementary row operations
math.la.c.mat.det.operations	math.la.d.mat.det.permutation	4	0.96
The permutation expansion for determinants
math.la.c.mat.det.operations	math.la.t.mat.det.transpose	4	0.95
A matrix and its transpose have the same determinant.
math.la.c.mat.det.operations	math.la.t.mat.det.product	4	0.94
If A and B are n-by-n matrices, then det(AB)=det(A)det(B).
math.la.c.mat.det.operations	math.la.t.mat.det.inv	4	0.93
The determinant of the inverse of A is the reciprocal of the determinant of A.
math.la.c.mat.det.operations	math.la.t.mat.det.block	4	0.92
The determinant of a block diagonal matrix is the product of the determinants of the blocks.
math.la.c.mat.det	math.la.c.mat.det.axiomatic	3	0.96
Determinants axiomatically
math.la.c.mat.det.axiomatic	math.la.d.multilinear	4	0.99
Definition of multilinear function
math.la.c.mat.det.axiomatic	math.la.t.mat.det.exists	4	0.98
The determinant function exists.
math.la.c.mat.det.axiomatic	math.la.t.mat.det.unique	4	0.97
The determinant function is unique.
math.la.c.mat.det.axiomatic	math.la.t.det.multilinear	4	0.96
A determinant is a multilinear function
math.la.c.mat.det.axiomatic	math.la.t.mat.det.add_mult	4	0.95
Adding a multiple of one row to another row does not change the determinant.
math.la.c.mat.det.axiomatic	math.la.t.mat.det.switch	4	0.94
Switching two rows multiplies the determinant by -1.
math.la.c.mat.det.axiomatic	math.la.t.mat.det.scalar	4	0.93
Multiplying a row by a scalar multiplies the determinant by that scalar.
math.la.c.mat.det.axiomatic	math.la.t.mat.row.equal	4	0.92
A matrix with two equal rows/columns has determinant 0
math.la.c.mat.det.axiomatic	math.la.t.mat.row.z	4	0.91
A matrix with a 0 row/column has determinant 0
math.la.c.mat.det	math.la.c.mat.det.types	3	0.95
Particular types of matrices
math.la.c.mat.det.types	math.la.t.mat.det.2x2	4	0.99
Formula for the determinant of a 2-by-2 matrix.
math.la.c.mat.det.types	math.la.t.mat.det.3x3	4	0.98
Formula for the determinant of a 3-by-3 matrix.
math.la.c.mat.det.types	math.la.t.mat.det.trianglar	4	0.97
The determinant of a triangular matrix is the product of the entries on the diagonal.
math.la.c.mat.det.types	math.la.t.mat.elementary.det	4	0.96
Theorem describing the determinants of elementary matrices.
math.la.c.mat.det	math.la.t.mat.det.col.volume	3	0.94
The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns.
math.la.c.mat.det	math.la.t.lintrans.det.volume	3	0.93
The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.
math.la.c.mat.det	math.la.d.mat.classicaladjoint	3	0.92
Definition of adjugate/classical adjoint of a matrix
math.la.c.mat.det	math.la.c.mat.illconditioned	3	0.91
A matrix is called ill-conditioned if it is nearly singular
math.la.c.mat.det	math.la.c.mat.conditionnumber	3	0.90
The condition number of matrix measures how close it is to being singular
math.la	math.la.c.linsys.mat	1	0.97
Linear systems and matrices
math.la.c.linsys.mat	math.la.t.linsys.vec	2	0.99
A linear system is equivalent to a vector equation.
math.la.c.linsys.mat	math.la.t.linsys.mat	2	0.98
A linear system is equivalent to a matrix equation.
math.la.c.linsys.mat	math.la.c.linsys.mat.terminology	2	0.97
Terminology
math.la.c.linsys.mat.terminology	math.la.d.mat.augmented	3	0.99
Definition of augmented matrix (of a linear system)
math.la.c.linsys.mat.terminology	math.la.d.mat.coeff	3	0.98
Definition of coefficient matrix of a linear system
math.la.c.linsys.mat.terminology	math.la.d.vec.constant	3	0.97
Definition of constant vector of a linear system
math.la.c.linsys.mat.terminology	math.la.d.vec.solution	3	0.96
Definition of solution vector of a linear system
math.la.c.linsys.mat.terminology	math.la.d.linsys.mat.repn	3	0.95
Definition of matrix representation of a linear system
math.la.c.linsys.mat	math.la.c.linsys.mat.solve	2	0.96
Using matrices to solve linear systems
math.la.c.linsys.mat.solve	math.la.e.linsys.3x3.soln.row_reduce.o	3	0.99
Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of one solution
math.la.c.linsys.mat.solve	math.la.e.linsys.3x3.soln.row_reduce.z	3	0.98
Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of no solutions
math.la.c.linsys.mat.solve	math.la.e.linsys.3x3.soln.row_reduce.i	3	0.97
Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions
math.la.c.linsys.mat.solve	math.la.e.linsys.3x3.soln.homog.row_reduce.o	3	0.96
Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of one solution
math.la.c.linsys.mat.solve	math.la.e.linsys.3x3.soln.homog.row_reduce.i	3	0.95
Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions
math.la.c.linsys.mat	math.la.c.mat.eqn	2	0.95
Matrix equations
math.la.c.mat.eqn	math.la.d.mat.eqn	3	0.99
Definition of matrix equation
math.la.c.mat.eqn	math.la.e.mat.eqn.3x3.solve	3	0.98
Example of solving a 3-by-3 matrix equation
math.la.c.mat.eqn	math.la.e.mat.eqn.3x3.homog.solve	3	0.97
Example of solving a 3-by-3 homogeneous matrix equation
math.la.c.mat.eqn	math.la.t.mat.eqn.linsys	3	0.96
A matrix equation is equivalent to a linear system
math.la.c.mat.eqn	math.la.t.mat.eqn.lincomb	3	0.95
The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.
math.la.c.linsys.mat	math.la.t.mat.row_equiv.linsys	2	0.94
Row equivalent matrices represent equivalent linear systems
math.la.c.linsys.mat	math.la.c.linsys.mat.echelon	2	0.93
Linear systems and echelon matrices
math.la.c.linsys.mat.echelon	math.la.t.echelon.consistent	3	0.99
The echelon form can be used to determine if a linear system is consistent.
math.la.c.linsys.mat.echelon	math.la.e.echelon.consistent	3	0.98
Example of using the echelon form to determine if a linear system is consistent.
math.la.c.linsys.mat.echelon	math.la.t.rref.pivot.oi	3	0.97
The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.
math.la.c.linsys.mat.echelon	math.la.t.rref.pivot.free	3	0.96
The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.
math.la	math.la.c.vsp	1	0.96
Vector spaces
math.la.c.vsp	math.la.c.vsp.coord	2	0.99
Coordinate vector spaces
math.la.c.vsp.coord	math.la.c.vec.rncn	3	0.99
Algebraic properties of R^n (or C^n)
math.la.c.vec.rncn	math.la.d.scalar.coord	4	0.99
Definition of scalar, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.coord	4	0.98
Definition of vector, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.col.coord	4	0.97
Definition of column vector, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.rncn	4	0.96
Definition of R^n (or C^n)
math.la.c.vec.rncn	math.la.d.vec.size.coord	4	0.95
Definition of size of a vector, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.component.coord	4	0.94
Definition of entry/component of a vector, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.z.coord	4	0.93
Definition of 0 vector, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.equal.coord	4	0.92
Definition of equality of vectors, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.sum.coord	4	0.91
Definition of vector sum/addition, coordinate vector space
math.la.c.vec.rncn	math.la.t.vec.sum.coord	4	0.90
Vector sum/addition is commutative and associative, coordinate vector space
math.la.c.vec.rncn	math.la.d.vec.conjugate.cn	4	0.89
Definition of conjugate of a vector in C^n
math.la.c.vec.rncn	math.la.d.vec.real.cn	4	0.88
Definition of the real part of a vector in C^n
math.la.c.vec.rncn	math.la.d.vec.imaginary.cn	4	0.87
Definition of the imaginary part of a vector in C^n
math.la.c.vec.rncn	math.la.t.vec.sum.conjugate.cn	4	0.86
The conjugate of a sum of vectors in C^n is the sum of the conjugates
math.la.c.vec.rncn	math.la.d.vec.scalar.mult.coord	4	0.85
Definition of vector-scalar multiplication, coordinate vector space
math.la.c.vec.rncn	math.la.t.vec.scalar.mult.conjugate.cn	4	0.84
The conjugate of vector-scalar multiplication in C^n is the product of the conjugates.
math.la.c.vec.rncn	math.la.e.vec.scalar.mult.r2	4	0.83
Example of vector-scalar multiplication in R^2
math.la.c.vsp.coord	math.la.c.vec.rncn.geom	3	0.98
Geometric properties of R^n (or C^n)
math.la.c.vec.rncn.geom	math.la.e.vec.sum.geometric.r2	4	0.99
Example of a sum of vectors interpreted geometrically in R^2
math.la.c.vec.rncn.geom	math.la.t.vec.sum.geometric.rncn	4	0.98
Vector sum/addition interpreted geometrically in R^n (or C^n)
math.la.c.vsp.coord	math.la.d.vsp.axioms.coord	3	0.97
Axioms of a vector space, coordinate vector space
math.la.c.vsp.coord	math.la.c.vec.lincomb.coord	3	0.96
Linear combinations
math.la.c.vec.lincomb.coord	math.la.d.vec.lincomb.coord	4	0.99
Definition of linear combination of vectors, coordinate vector space
math.la.c.vec.lincomb.coord	math.la.e.vec.lincomb.r2	4	0.98
Example of linear combination of vectors in R^2
math.la.c.vec.lincomb.coord	math.la.d.vec.lincomb.weight.coord	4	0.97
Definition of weights in a linear combination of vectors, coordinate vector space
math.la.c.vec.lincomb.coord	math.la.e.vec.lincomb.weight.solve.r3	4	0.96
Example of writing a given vector in R^3 as a linear combination of given vectors
math.la.c.vsp.coord	math.la.c.vec.span.coord	3	0.95
Spans
math.la.c.vec.span.coord	math.la.d.vec.span.coord	4	0.99
Definition of span of a set of vectors, coordinate vector space
math.la.c.vec.span.coord	math.la.c.vec.span.geometric.rncn	4	0.98
Geometric description of span of a set of vectors in R^n (or C^n)
math.la.c.vec.span.coord	math.la.e.vec.span.r3	4	0.97
Determine if a particular set of vectors spans R^3
math.la.c.vec.span.coord	math.la.e.vec.span.of	4	0.96
Determine if a particular vector is in the span of a set of vectors
math.la.c.vec.span.coord	math.la.e.vec.span.of.r2	4	0.95
Determine if a particular vector is in the span of a set of vectors in R^2
math.la.c.vec.span.coord	math.la.e.vec.span.of.r3	4	0.94
Determine if a particular vector is in the span of a set of vectors in R^3
math.la.c.vsp.coord	math.la.c.vsp.subspace.coord	3	0.94
Subspaces
math.la.c.vsp.subspace.coord	math.la.d.vsp.subspace.coord	4	0.99
Definition of subspace, coordinate vector space
math.la.c.vsp.subspace.coord	math.la.d.vec.span.subspace.coord	4	0.98
Definition of subspace spanned by a set of a set of vectors, coordinate vector space
math.la.c.vsp.subspace.coord	math.la.d.vsp.subspace.z.coord	4	0.97
Definition of the 0/trivial subspace
math.la.c.vsp.coord	math.la.c.vec.lindep	3	0.93
Linear (in)dependence
math.la.c.vec.lindep	math.la.d.vec.lindep.relation	4	0.99
Definition of linear dependence relation on a set of vectors
math.la.c.vec.lindep	math.la.d.vec.lindep.relation.trivial	4	0.98
Definition of trivial linear dependence relation on a set of vectors
math.la.c.vec.lindep	math.la.e.vec.linindep.r3	4	0.97
Determine if a particular set of vectors in R^3 in linearly independent
math.la.c.vec.lindep	math.la.d.vec.linindep.coord	4	0.96
Definition of linearly independent set of vectors: if a linear combination is 0, then every coefficient is 0, coordinate vector space.
math.la.c.vec.lindep	math.la.t.vec.lindep.coord	4	0.95
Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space.
math.la.c.vec.lindep	math.la.d.vec.lindep.coord	4	0.94
Definition of linearly dependent set of vectors:  one of the vectors can be written as a linear combination of the other vectors, coordinate vector space.
math.la.c.vec.lindep	math.la.t.vec.linindep.coord	4	0.93
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0, then every coefficient is 0, coordinate vector space.
math.la.c.vec.lindep	math.la.t.vec.linindep.homog	4	0.92
A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.
math.la.c.vec.lindep	math.la.t.vec.linindep.pivot	4	0.91
A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.
math.la.c.vec.lindep	math.la.t.vec.lindep.z	4	0.90
If a set of vectors contains the 0 vector, then the set is linearly dependent.
math.la.c.vec.lindep	math.la.t.vec.lindep.two	4	0.89
A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.
math.la.c.vec.lindep	math.la.t.vec.lindep.more.rncn	4	0.88
If a set of vectors in R^n (or C^n) contains more than n elements, then the set is linearly dependent.
math.la.c.vsp.coord	math.la.c.vsp.basis.coord	3	0.92
Bases
math.la.c.vsp.basis.coord	math.la.d.vsp.basis.coord	4	0.99
Definition of basis of a vector space (or subspace), coordinate vector space
math.la.c.vsp.basis.coord	math.la.d.vsp.basis.standard.rncn	4	0.98
Definition of the standard/natural basis of R^n (or C^n)
math.la.c.vsp.basis.coord	math.la.t.vsp.basis.standard.rncn	4	0.97
The standard/natural basis of R^n (or C^n) is a basis.
math.la.c.vsp.basis.coord	math.la.d.vsp.basis.coord.change.rncn	4	0.96
Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)
math.la.c.vsp.basis.coord	math.la.d.vsp.basis.standard.leq_n	4	0.95
Definition of the standard basis of the polynomials of degree at most n
math.la.c.vsp.basis.coord	math.la.d.vsp.basis.standard.m_by_n	4	0.94
Definition of the standard basis of the m by n matrices
math.la.c.vsp.basis.coord	math.la.d.vsp.basis.relative.coord	4	0.93
Definition of coordinates relative to a given basis, coordinate vector space
math.la.c.vsp.coord	math.la.c.vsp.dim.coord	3	0.91
Dimension
math.la.c.vsp.dim.coord	math.la.d.vsp.dim.coord	4	0.99
Definition of dimension of a vector space (or subspace), coordinate vector space
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.span.coord	4	0.98
If a vector space has dimension n, then any subset set of n vectors that spans the space must be a basis, coordinate vector space.
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.linindep.coord	4	0.97
If a vector space has dimension n, then any subset of n vectors that is linearly independent must be a basis, coordinate vector space.
math.la.c.vsp.coord	math.la.d.lintrans.coord	3	0.90
Linear transformations
math.la.d.lintrans.coord	math.la.c.lintrans.basic.coord	4	0.99
Terminology
math.la.c.lintrans.basic.coord	math.la.d.lintrans.coord	5	0.99
Definition of linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.domain.coord	5	0.98
Definition of domain of a linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.codomain.coord	5	0.97
Definition of codomain of a linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.image.coord	5	0.96
Definition of image (of a point) under a linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.preimage.coord	5	0.95
Definition of pre-image (of a point) under a linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.surjective.coord	5	0.94
Definition of onto/surjective linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.injective.coord	5	0.93
Definition of one-to-one/injective linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.range.coord	5	0.92
Definition of range of linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.kernel.coord	5	0.91
Definition of kernel of linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.invertible.coord	5	0.90
Definition of invertible linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.d.lintrans.inv.coord	5	0.89
Definition of inverse of a linear transformation, coordinate vector space
math.la.c.lintrans.basic.coord	math.la.e.lintrans.not	5	0.88
Non-example of a linear transformation
math.la.d.lintrans.coord	math.la.c.lintrans.geometric	4	0.98
Geometric properties of linear transformations
math.la.c.lintrans.geometric	math.la.c.lintrans.r2.region	5	0.99
Visualise a linear transformation on R^2 by looking at the image of a region
math.la.c.lintrans.geometric	math.la.c.lintrans.geometric.r2	5	0.98
Geometric properties of linear transformations on R^2
math.la.d.lintrans.coord	math.la.c.mat.transformation	4	0.97
Matrices as linear transformations
math.la.c.mat.transformation	math.la.c.transformation.matrix	5	0.99
Matrices act as a transformation by multiplying vectors
math.la.c.mat.transformation	math.la.t.mat.vec.mult.lintrans	5	0.98
Matrix-vector multiplication is a linear transformation.
math.la.c.mat.transformation	math.la.d.lintrans.mat.basis.standard.coord	5	0.97
Definition of the standard matrix for a linear transformation, coordinate setting
math.la.c.mat.transformation	math.la.t.lintrans.mat.basis.standard.coord	5	0.96
A linear transformation is given by a matrix whose columns are the images of the standard basis vectors, coordinate setting.
math.la.c.mat.transformation	math.la.t.lintrans.mat.basis	5	0.95
A linear transformation is given by a matrix with respect to a given basis.
math.la.c.mat.transformation	math.la.d.lintrans.nilpotent	5	0.94
Definition of nilpotent linear transformation
math.la.c.mat.transformation	math.la.d.lintrans.mat.repn.coord	5	0.93
Definition of matrix representation of a linear transformation, coordinate vector space
math.la.c.mat.transformation	math.la.t.lintrans.mat_repn.composition.coord	5	0.92
Matrix representation of a composition of linear transformations is given by a matrix product, coordinate vector space
math.la.d.lintrans.coord	math.la.c.lintrans.basic	4	0.96
Basic properties of linear transformations
math.la.c.lintrans.basic	math.la.t.lintrans.z	5	0.99
A linear transformation maps 0 to 0.
math.la.c.lintrans.basic	math.la.t.lintrans.lincomb	5	0.98
A linear transformation of a linear combination is the linear combination of the linear transformation
math.la.c.lintrans.basic	math.la.t.lintrans.basis	5	0.97
A linear transformation is determined by its action on a basis.
math.la.c.lintrans.basic	math.la.d.lintrans.basis.extension	5	0.96
Definition of how the action of a linear transformation on a basis extends to the whole space
math.la.c.lintrans.basic	math.la.t.linsys.homog.soln_preimage	5	0.95
The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.
math.la.d.lintrans.coord	math.la.t.mat.null_space.rref.span	4	0.95
Description of a spanning set for the null space of a matrix from the reduced row-echelon form.
math.la.d.lintrans.coord	math.la.t.mat.null_space.rref.basis	4	0.94
Description of a basis for the null space of a matrix from the reduced row-echelon form.
math.la.d.lintrans.coord	math.la.d.mat.echelon.linindep	4	0.93
The nonzero rows of an echelon form of a matrix are linearly independent.
math.la.d.lintrans.coord	math.la.c.mat.subspace	4	0.92
Subspaces associated to a matrix
math.la.c.mat.subspace	math.la.d.mat.null_space.right	5	0.99
Definition of matrix null space (right)
math.la.c.mat.subspace	math.la.d.mat.null_space.left	5	0.98
Definition of matrix null space (left)
math.la.c.mat.subspace	math.la.d.mat.col_space	5	0.97
Definition of column space of a matrix
math.la.c.mat.subspace	math.la.t.mat.col_space.rncn	5	0.96
The column space of an m-by-n matrix is a subspace of R^m (or C^m)
math.la.c.mat.subspace	math.la.t.mat.col_space.pivot	5	0.95
The pivot columns of a matrix are a basis for the column space.
math.la.c.mat.subspace	math.la.c.mat.col_space.row_reduce	5	0.94
Row operations do not necessarily preserve the column space.
math.la.c.mat.subspace	math.la.t.mat.row_space.pivot	5	0.93
The nonzero rows of the reduced row-echelon form of a matris are a basis for the row space.
math.la.c.mat.subspace	math.la.d.mat.row_space	5	0.92
Definition of row space of a matrix
math.la.c.mat.subspace	math.la.d.mat.row_space.row_equiv	5	0.91
Row equivalent matrices have the same row space.
math.la.d.lintrans.coord	math.la.c.lintrans.rank	4	0.91
Rank and nullity
math.la.c.lintrans.rank	math.la.d.lintrans.rank	5	0.99
Definition of rank of a linear transformation
math.la.c.lintrans.rank	math.la.d.lintrans.nullity	5	0.98
Definition of nullity of a linear transformation
math.la.c.lintrans.rank	math.la.t.lintrans.surjective.span	5	0.97
A linear transformation is surjective if and only if the columns of its matrix span the codomain.
math.la.d.lintrans.coord	math.la.c.litrans.example	4	0.90
Examples
math.la.c.litrans.example	math.la.t.mat.rotation	5	0.99
Matrix describing a rotation of the plane
math.la.c.litrans.example	math.la.e.lintrans.generic.r2	5	0.98
Example of a linear transformation on R^2: generic
math.la.c.litrans.example	math.la.e.lintrans.shear.r2	5	0.97
Example of a linear transformation on R^2: shear
math.la.c.litrans.example	math.la.e.lintrans.rotation.r2	5	0.96
Example of a linear transformation on R^2: rotation
math.la.c.litrans.example	math.la.e.lintrans.projection.r2	5	0.95
Example of a linear transformation on R^2: projection
math.la.c.litrans.example	math.la.e.lintrans.rotation.r3	5	0.94
Example of a linear transformation on R^3: rotation
math.la.c.vsp.coord	math.la.c.vec.orthog_proj.coord	3	0.89
Orthogonality and projection
math.la.c.vec.orthog_proj.coord	math.la.c.innerproduct.coord	4	0.99
Inner products in coordinate spaces
math.la.c.innerproduct.coord	math.la.d.innerproduct.rn	5	0.99
Definition of inner/dot product on R^n, in terms of coordinates
math.la.c.innerproduct.coord	math.la.d.innerproduct.cn	5	0.98
Definition of inner/dot product on C^n, in terms of coordinates
math.la.c.innerproduct.coord	math.la.t.innerproduct.commutative.rn	5	0.97
The standard inner product on R^n is commutative.
math.la.c.innerproduct.coord	math.la.t.innerproduct.commutative.cn	5	0.96
The standard inner product on C^n is anticommutative.
math.la.c.innerproduct.coord	math.la.t.innerproduct.commutative.scalar.rn	5	0.95
The standard inner product on R^n commutes with (real) scalar multiplication.
math.la.c.innerproduct.coord	math.la.t.innerproduct.commutative.scalar.cn	5	0.94
The standard inner product on C^n commutes/anticommutes with scalar multiplication.
math.la.c.innerproduct.coord	math.la.t.innerproduct.distributive.rncn	5	0.93
The standard inner product on R^n (or C^n) distributes over addition.
math.la.c.innerproduct.coord	math.la.t.innerproduct.self.positive.coord	5	0.92
The standard inner product of a vector with itself is non-negative, coordinate setting.
math.la.c.innerproduct.coord	math.la.t.innerproduct.self.z.coord	5	0.91
The standard inner product of a vector with itself is 0 only for the 0 vector, coordinate setting.
math.la.c.innerproduct.coord	math.la.t.innerproduct.mat.rn	5	0.90
The standard inner product on R^n can be written as the product of a vector and the transpose of a vector.
math.la.c.innerproduct.coord	math.la.t.innerproduct.mat.cn	5	0.89
The standard inner product on C^n can be written as the product of a vector and the adjoint of a vector.
math.la.c.innerproduct.coord	math.la.t.innerproduct.adjoint.cn	5	0.88
A matrix turns into its adjoint when moved to the other side of the standard inner product on C^n.
math.la.c.vec.orthog_proj.coord	math.la.c.vec.norm.coord	4	0.98
Norm and length
math.la.c.vec.norm.coord	math.la.d.vec.norm.coord	5	0.99
Definition of norm/length of a vector, coordinate setting
math.la.c.vec.norm.coord	math.la.d.vec.unit.coord	5	0.98
Definition of unit vector, coordinate setting
math.la.c.vec.norm.coord	math.la.t.vec.innerproduct.norm	5	0.97
The inner product of a vector with itself is the square of its norm/length.
math.la.c.vec.norm.coord	math.la.d.distance.coord	5	0.96
Definition of distance, coordinate setting
math.la.c.vec.norm.coord	math.la.t.vec.triangle.coord	5	0.95
The triangle inequality, coordinate setting
math.la.c.vec.norm.coord	math.la.t.vec.cauchyschwartz.coord	5	0.94
The Cauchy-Schwartz inequality, coordinate setting
math.la.c.vec.orthog_proj.coord	math.la.c.vec.orthogonal.coord	4	0.97
Orthogonality
math.la.c.vec.orthogonal.coord	math.la.d.vec.angle.coord	5	0.99
Definition of angle between vectors, coordinate setting
math.la.c.vec.orthogonal.coord	math.la.d.vec.orthogonal.coord	5	0.98
Definition of orthogonal vectors, coordinate setting
math.la.c.vec.orthogonal.coord	math.la.d.vec.parallel.coord	5	0.97
Definition of parallel vectors, coordinate setting
math.la.c.vec.orthogonal.coord	math.la.t.vec.orthogonal	5	0.96
Two vectors are orthogonal if and only if the Pythagorean Theorem holds.
math.la.c.vec.orthogonal.coord	math.la.d.vec.subspace.orthogonal	5	0.95
Definition of a vector being orthogonal to a subspace
math.la.c.vec.orthogonal.coord	math.la.d.subspace.orthogonal_complement	5	0.94
Definition of orthogonal complement of a subspace
math.la.c.vec.orthogonal.coord	math.la.t.subspace.orthogonal_complement	5	0.93
The orthogonal complement of a subspace is a subspace.
math.la.c.vec.orthogonal.coord	math.la.t.subspace.orthogonal_complement.sum	5	0.92
The direct sum of a subspace and its orthogonal complement is the whole space.
math.la.c.vec.orthogonal.coord	math.la.t.subspace.orthogonal_complement.basis	5	0.91
A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.
math.la.c.vec.orthogonal.coord	math.la.t.mat.row.null.orthogonal_complement	5	0.90
The null space of a matrix is the orthogonal complement of the column space.
math.la.c.vec.orthogonal.coord	math.la.d.vec.orthogonal_set	5	0.89
Definition of orthogonal set of vectors
math.la.c.vec.orthogonal.coord	math.la.d.vec.orthonormal_set	5	0.88
Definition of orthonormal set of vectors
math.la.c.vec.orthogonal.coord	math.la.t.vec.orthogonal_set.linindep	5	0.87
An orthogonal set of nonzero vectors is linearly independent.
math.la.c.vec.orthogonal.coord	math.la.d.subspace.basis.orthogonal	5	0.86
Definition of orthogonal basis of a (sub)space
math.la.c.vec.orthogonal.coord	math.la.d.subspace.basis.orthonormal	5	0.85
Definition of orthonormal basis of a (sub)space
math.la.c.vec.orthogonal.coord	math.la.t.mat.col.orthonormal.inv.rn	5	0.84
A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose.
math.la.c.vec.orthogonal.coord	math.la.t.mat.col.orthonormal.norm.rn	5	0.83
A matrix with real entries and orthonormal columns preserves norms.
math.la.c.vec.orthogonal.coord	math.la.t.mat.col.orthonormal.dot.rn	5	0.82
A matrix with real entries and orthonormal columns preserves dot products.
math.la.c.vec.orthogonal.coord	math.la.t.subspace.basis.orthogonal	5	0.81
Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.
math.la.c.vec.orthogonal.coord	math.la.t.vec.projection.subspace	5	0.80
A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.
math.la.c.vec.orthogonal.coord	math.la.d.gramschmidt	5	0.79
Description of the Gram-Schmidt process
math.la.c.vec.orthogonal.coord	math.la.t.gramschmidt	5	0.78
The Gram-Schmidt process converts a linearly independent set into an orthogonal set.
math.la.c.vec.orthog_proj.coord	math.la.c.vec.proj.coord	4	0.96
Projection
math.la.c.vec.proj.coord	math.la.d.vec.projection	5	0.99
Definition of (orthogonal) projection of one vector onto another vector
math.la.c.vec.proj.coord	math.la.t.vec.projection	5	0.98
Formula for the (orthogonal) projection of one vector onto another vector
math.la.c.vec.proj.coord	math.la.d.vec.projection_arbitrary.subspace	5	0.97
Definition of (not necessarily orthogonal) projection onto a component of a direct sum
math.la.c.vec.proj.coord	math.la.d.vec.projection.subspace	5	0.96
Definition of (orthogonal) projection onto a subspace
math.la.c.vec.proj.coord	math.la.t.vec.projection.element	5	0.95
The projection of a vector which is in a subspace is the vector itself.
math.la.c.vec.proj.coord	math.la.t.vec.projection.closest	5	0.94
The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector.
math.la.c.vec.proj.coord	math.la.t.subspace.basis.orthonormal	5	0.93
Formula for the coordinates of the projection of a vector onto a subspace, with respect to an orthonormal basis.
math.la.c.vec.orthog_proj.coord	math.la.c.linsys.leastsquares	4	0.95
Least squares
math.la.c.linsys.leastsquares	math.la.d.linsys.leastsquares	5	0.99
Definition of least-squares solution to a linear system
math.la.c.linsys.leastsquares	math.la.d.linsys.leastsquares.error	5	0.98
Definition of least-squares error of a linear system
math.la.c.linsys.leastsquares	math.la.t.linsys.leastsquares	5	0.97
Formula for computing the least squares solution to a linear system.
math.la.c.linsys.leastsquares	math.la.t.linsys.leastsquares.qr	5	0.96
Formula for computing the least squares solution to a linear system, in terms of the QR factorization of the coefficient matrix.
math.la.c.linsys.leastsquares	math.la.t.linsys.leastsquares.unique	5	0.95
The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.
math.la.c.linsys.leastsquares	math.la.d.leastsquares.line	5	0.94
Definition of the least-squares linear fit to 2-dimensional data
math.la.c.linsys.leastsquares	math.la.t.leastsquares.line	5	0.93
Formula for the least-squares linear fit to 2-dimensional data
math.la.c.vsp	math.la.c.vsp.abs	2	0.98
Abstract vector spaces
math.la.c.vsp.abs	math.la.c.vsp.abs.defn	3	0.99
Definition and terminology
math.la.c.vsp.abs.defn	math.la.d.vec.arb	4	0.99
Definition of vector, arbitrary vector space
math.la.c.vsp.abs.defn	math.la.d.scalar.arb	4	0.98
Definition of scalar, arbitrary vector space
math.la.c.vsp.abs.defn	math.la.d.vec.add.arb	4	0.97
Definition of vector addition, arbitrary vector space
math.la.c.vsp.abs.defn	math.la.d.vec.scalar.mult.arb	4	0.96
Definition of vector-scalar multiplication, arbitrary vector space
math.la.c.vsp.abs.defn	math.la.d.vsp.axioms.arb	4	0.95
Axioms of a vector space, arbitrary vector space
math.la.c.vsp.abs.defn	math.la.d.vsp.vector.negative	4	0.94
The additive inverse of a vector is called the negative of the vector.
math.la.c.vsp.abs	math.la.c.vsp.abs.basic	3	0.98
Basic properties
math.la.c.vsp.abs.basic	math.la.t.vsp.z.unique	4	0.99
The 0 vector is unique.
math.la.c.vsp.abs.basic	math.la.d.vsp.vector.negative.unique	4	0.98
The additive inverse of a vector is unique.
math.la.c.vsp.abs.basic	math.la.t.vsp.scalar.mult.z	4	0.97
The 0 scalar multiplied by any vector equals the 0 vector.
math.la.c.vsp.abs.basic	math.la.t.vsp.vector.mult.z	4	0.96
The 0 vector multiplied by any scalar equals the 0 vector.
math.la.c.vsp.abs.basic	math.la.t.vsp.mult.z	4	0.95
If the product of a vector and a scalar is 0, then one of them is 0.
math.la.c.vsp.abs.basic	math.la.t.vsp.vector.negative	4	0.94
The additive inverse of a vector equals the vector multiplied by -1.
math.la.c.vsp.abs	math.la.c.vsp.subspace.arb	3	0.97
Subspaces
math.la.c.vsp.subspace.arb	math.la.d.vsp.subspace.arb	4	0.99
Definition of subspace, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.d.vsp.subspace.z.arb	4	0.98
Definition of 0/trivial subspace, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.t.vsp.subspace.lincomb.arb	4	0.97
A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.d.vsp.subspace.intersection.arb	4	0.96
Definition of intersection of subspaces, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.t.vsp.subspace.intersection.arb	4	0.95
The intersection of subspaces is a subspace, arbitrary vector space.
math.la.c.vsp.subspace.arb	math.la.d.vsp.subspace.sum.arb	4	0.94
Definition of sum of subspaces, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.t.vsp.subspace.sum.arb	4	0.93
The sum of subspaces is a subspace, arbitrary vector space.
math.la.c.vsp.subspace.arb	math.la.d.vsp.subspace.sum.direct.arb	4	0.92
Definition of direct sum of subspaces, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.t.vsp.subspace.sum.direct.dim	4	0.91
The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.
math.la.c.vsp.subspace.arb	math.la.d.vsp.subspace.independent.arb	4	0.90
Definition of independent subspaces, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.t.vsp.subspace.independent.lincomb	4	0.89
A vector can be written uniquely as a linear combination of vectors from independent subspaces.
math.la.c.vsp.subspace.arb	math.la.t.vsp.subspace.independent.basis	4	0.88
The union of bases from independent subspaces is a basis for the space.
math.la.c.vsp.subspace.arb	math.la.d.vsp.subspace.complement.arb	4	0.87
Definition of complement of a subspace, arbitrary vector space
math.la.c.vsp.subspace.arb	math.la.t.vsp.subspace.complement.arb	4	0.86
Theorem characterizing when a space is the direct sum of two subspaces
math.la.c.vsp.abs	math.la.c.vec.lincomb.abs	3	0.96
Linear combinations
math.la.c.vec.lincomb.abs	math.la.d.vec.lincomb.arb	4	0.99
Definition of linear combination of vectors, arbitrary vector space
math.la.c.vsp.abs	math.la.c.vec.span.coord	3	0.95
Spans
math.la.c.vec.span.coord	math.la.d.vec.span.arb	4	0.99
Definition of span of a set of vectors, arbitrary vector space
math.la.c.vec.span.coord	math.la.t.vec.span.subspace.arb	4	0.98
The span of a set of vectors is a subspace, arbitrary vector space
math.la.c.vec.span.coord	math.la.d.vsp.span.set.arb	4	0.97
Definition of spanning/generating set for a space or subspace, arbitrary vector space
math.la.d.vsp.span.set.arb	math.la.t.vsp.dim.less.span.arb	5	0.99
A set of vectors containing fewer elements than the dimension of the space cannot span, arbitrary vector space.
math.la.c.vsp.abs	math.la.c.vec.lindep.arb	3	0.94
Linear (in)dependence
math.la.c.vec.lindep.arb	math.la.d.vec.linindep.arb	4	0.99
Definition of linearly independent set of vectors: if a linear combination is 0, then every coefficient is 0, arbitrary vector space.
math.la.c.vec.lindep.arb	math.la.t.vec.lindep.arb	4	0.98
Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, arbitrary vector space.
math.la.c.vec.lindep.arb	math.la.d.vec.lindep.arb	4	0.97
Definition of linearly dependent set of vectors:  one of the vectors can be written as a linear combination of the other vectors, arbitrary vector space.
math.la.c.vec.lindep.arb	math.la.t.vec.linindep.arb	4	0.96
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0, then every coefficient is 0, arbitrary vector space.
math.la.c.vec.lindep.arb	math.la.t.vsp.linindep.subset	4	0.95
A subset of a linearly independent set is linearly independent.
math.la.c.vec.lindep.arb	math.la.t.vsp.linindep.coord	4	0.94
A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.
math.la.c.vec.lindep.arb	math.la.t.vsp.span.lindep	4	0.93
Removing a linearly dependent vector from a set does not change the span of the set.
math.la.c.vec.lindep.arb	math.la.t.vsp.linindep.extend	4	0.92
Adjoining an element not in the span of a linearly independent set gives another linearly independent set.
math.la.c.vec.lindep.arb	math.la.t.vsp.linindep.basis.arb	4	0.91
Any linearly independent set can be expanded to a basis for the (sub)space, arbitrary vector space.
math.la.c.vsp.abs	math.la.c.vsp.basis.arb	3	0.93
Bases
math.la.c.vsp.basis.arb	math.la.d.vsp.basis.arb	4	0.99
Definition of basis of a vector space (or subspace), arbitrary vector space
math.la.c.vsp.basis.arb	math.la.t.vsp.span.basis	4	0.98
A set of nonzero vectors contains (as a subset) a basis for its span.
math.la.c.vsp.basis.arb	math.la.t.vsp.span.basis.rref	4	0.97
The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.
math.la.c.vsp.basis.arb	math.la.t.vsp.basis.coord.unique	4	0.96
Each vector can be written uniquely as a linear combination of vectors from a given basis.
math.la.c.vsp.basis.arb	math.la.t.vsp.basis.span.unique	4	0.95
A set is a basis if each vector can be written uniquely as a linear combination.
math.la.c.vsp.basis.arb	math.la.c.vsp.coord.arb	4	0.94
Coordinates
math.la.c.vsp.coord.arb	math.la.d.vsp.basis.relative.arb	5	0.99
Definition of coordinates relative to a given basis, arbitrary vector space
math.la.c.vsp.coord.arb	math.la.d.vsp.basis.exchange.arb	5	0.98
If B is a basis containing b and the b coordinate of c is nonzero, the replacing b with c gives another basis.
math.la.c.vsp.coord.arb	math.la.d.vsp.basis.coord.vector.arb	5	0.97
Definition of coordinate vector/mapping/representation relative to a given basis, arbitrary vector space
math.la.c.vsp.coord.arb	math.la.t.vsp.basis.coord.lin.arb	5	0.96
The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).
math.la.c.vsp.coord.arb	math.la.t.vsp.basis.coord.injective.arb	5	0.95
The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).
math.la.c.vsp.coord.arb	math.la.t.vsp.basis.coord.surjective.arb	5	0.94
The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).
math.la.c.vsp.coord.arb	math.la.d.vsp.change_of_basis.arb	5	0.93
Definition of change of coordinates matrix between two bases, arbitrary vector space
math.la.c.vsp.coord.arb	math.la.t.vsp.change_of_basis.exists.arb	5	0.92
The change of coordinates matrix between two bases exists and is unique, arbitrary vector space.
math.la.c.vsp.coord.arb	math.la.t.vsp.change_of_basis	5	0.91
Multiplication by a change of coordinates matrix converts representations for different bases.
math.la.c.vsp.coord.arb	math.la.t.vsp.change_of_basis.inv	5	0.90
Change of coordinates matrices are invertible, and the inverse changes coordinates in the other direction.
math.la.c.vsp.coord.arb	math.la.t.vsp.change_of_basis.conjugate	5	0.89
Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.
math.la.c.vsp.abs	math.la.c.vsp.isomorphism	3	0.92
Isomorphism
math.la.c.vsp.isomorphism	math.la.d.vsp.isomorphism	4	0.99
Definition of isomorphic/isomorphism between vector spaces
math.la.c.vsp.isomorphism	math.la.t.vsp.isomorphism.inv	4	0.98
The inverse of an isomorphism is an isomorphism.
math.la.c.vsp.isomorphism	math.la.t.vsp.isomorphism.equiv	4	0.97
Vector space isomorphism is an equivalence relation.
math.la.c.vsp.isomorphism	math.la.t.vsp.isomorphic.dim	4	0.96
Isomorphic vector spaces have the same dimension.
math.la.c.vsp.isomorphism	math.la.t.vsp.dim.isomorphic	4	0.95
Vector spaces with the same dimension are isomprphic.
math.la.c.vsp.isomorphism	math.la.t.vsp.isomorphic.rncn	4	0.94
Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.
math.la.c.vsp.isomorphism	math.la.d.vsp.automorphism	4	0.93
Definition of automorphism of a vector space
math.la.c.vsp.abs	math.la.c.vsp.dim.coord	3	0.91
Dimension
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.arb	4	0.99
Every basis for a vector space contains the same number of elements, arbitrary vector space.
math.la.c.vsp.dim.coord	math.la.d.vsp.dim.finite_infinite.arb	4	0.98
Definition of dimension of a vector space (or subspace) being finite or infinite, arbitrary vector space
math.la.c.vsp.dim.coord	math.la.d.vsp.dim.arb	4	0.97
Definition of dimension of a vector space (or subspace), arbitrary vector space
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.arb	4	0.96
The dimension of a vector space (or subspace) is well-defined, arbitrary vector space.
math.la.c.vsp.dim.coord	math.la.t.vsp.subspace.dim.arb	4	0.95
The dimension of a subspace is less than or equal to the dimension of the whole space, arbitrary vector space.
math.la.c.vsp.dim.coord	math.la.t.vsp.subspace.dim.equal	4	0.94
If two finite dimensional subspaces have the same dimension and one is contained in the other, then they are equal.
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.span.linindep.arb	4	0.93
If a vector space has dimension n, then any subset set of n vectors is linearly independent if and only if it spans, arbitrary vector space.
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.span.arb	4	0.92
If a vector space has dimension n, then any subset set of n vectors that spans the space must be a basis, arbitrary vector space.
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.linindep.arb	4	0.91
If a vector space has dimension n, then any subset of n vectors that is linearly independent must be a basis, arbitrary vector space.
math.la.c.vsp.dim.coord	math.la.t.vsp.dim.more.lindep.arb	4	0.90
A set of vectors containing more elements than the dimension of the space must be linearly dependent, arbitrary vector space.
math.la.c.vsp.abs	math.la.c.lintrans.arb	3	0.90
Linear transformations
math.la.c.lintrans.arb	math.la.c.lintrans.basic.arb	4	0.99
Terminology
math.la.c.lintrans.basic.arb	math.la.d.lintrans.arb	5	0.99
Definition of linear transformation/homomorphism, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.d.lintrans.identity	5	0.98
Definition of identity linear transformation
math.la.c.lintrans.basic.arb	math.la.d.lintrans.sum.arb	5	0.97
Definition of sum of linear transformations, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.t.lintrans.sum.arb	5	0.96
The sum of linear transformations is a linear transformation, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.d.lintrans.scalar.arb	5	0.95
Definition of scalar multiple of a linear transformation, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.t.lintrans.scalar.arb	5	0.94
A scalar multiple of a linear transformation is a linear transformation, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.d.lintrans.preimage.arb	5	0.93
Definition of pre-image of linear transformation, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.d.lintrans.surjective.arb	5	0.92
Definition of onto/surjective linear transformation, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.d.lintrans.range.arb	5	0.91
Definition of range of a linear transformation, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.d.lintrans.invertible.arb	5	0.90
Definition of invertible/nonsingular linear transformation, arbitrary vector space
math.la.c.lintrans.basic.arb	math.la.d.lintrans.inv.arb	5	0.89
Definition of inverse of a linear transformation, arbitrary vector space
math.la.c.lintrans.arb	math.la.c.lintrans.composition.arb	4	0.98
Composition
math.la.c.lintrans.composition.arb	math.la.d.lintrans.composition.arb	5	0.99
Definition of composition of linear transformations, arbitrary vector space.
math.la.c.lintrans.composition.arb	math.la.t.lintrans.composition.arb	5	0.98
The composition of linear transformations is a linear transformation, arbitrary vector space.
math.la.c.lintrans.composition.arb	math.la.t.lintrans.composition.injective.arb	5	0.97
The composition of injective linear transformations is injective, arbitrary vector space.
math.la.c.lintrans.composition.arb	math.la.t.lintrans.composition.surjective.arb	5	0.96
The composition of surjective linear transformations is surjective, arbitrary vector space.
math.la.c.lintrans.composition.arb	math.la.t.lintrans.composition.invertible.arb	5	0.95
The composition of invertible linear transformations is invertible, arbitrary vector space.
math.la.c.lintrans.composition.arb	math.la.t.lintrans.inv.arb	5	0.94
The inverse of a linear transformation is a linear transformation, arbitrary vector space
math.la.c.lintrans.composition.arb	math.la.t.lintrans.inv.involution.arb	5	0.93
The inverse of the inverse of a linear transformation is the original linear transformation, arbitrary vector space
math.la.c.lintrans.composition.arb	math.la.t.lintrans.inv.shoesandsocks	5	0.92
For invertible linear transformations A and B, the composition AB is invertible, and (AB)^-1=B^-1 A^-1.
math.la.c.lintrans.arb	math.la.t.lintrans.preimage.translation.arb	4	0.97
The preimage of a vector is a translation of the kernel of the linear transformation, arbitrary vector space
math.la.c.lintrans.arb	math.la.t.lintrans.injective.linindep	4	0.96
The image of a linearly independent set under an injective linear transformation is linearly independent.
math.la.c.lintrans.arb	math.la.t.lintrans.injective.dim	4	0.95
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
math.la.c.lintrans.arb	math.la.t.lintrans.surjective.dim	4	0.94
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
math.la.c.lintrans.arb	math.la.t.lintrans.surjective.rank	4	0.93
A linear transformation is surjective if and only if the rank equals the dimension of the codomain.
math.la.c.lintrans.arb	math.la.t.lintrans.range.subsp.arb	4	0.92
The range of a linear transformation is a subspace, arbitrary vector space
math.la.c.lintrans.arb	math.la.t.lintrans.range.span.arb	4	0.91
The the image of a spanning set is a spanning set for the range space, arbitrary vector space
math.la.c.lintrans.arb	math.la.t.lintrans.basis.span.surjective.arb	4	0.90
A linear transformation is surjective if and only if the image of a basis is a spanning set, arbitrary vector space
math.la.c.lintrans.arb	math.la.d.lintrans.range.generalized.arb	4	0.89
Definition of generalized range space of a linear transformation, arbitrary vector space
math.la.c.lintrans.arb	math.la.d.lintrans.range.generalized.injective	4	0.88
A linear transformation is injective on its generalized range space.
math.la.c.lintrans.arb	math.la.d.lintrans.diagonalizable	4	0.87
Definition of diagonalizable linear transformation
math.la.c.lintrans.arb	math.la.c.lintrans.eig.arb	4	0.86
Eigenvalues and eigenvectors
math.la.c.lintrans.eig.arb	math.la.d.lintrans.eig	5	0.99
Definition of eigenvalue/characteristic value of a linear transformation
math.la.c.lintrans.eig.arb	math.la.d.lintrans.eigvec	5	0.98
Definition of eigenvector/characteristic vector of a linear transformation
math.la.c.lintrans.eig.arb	math.la.d.lintrans.charpoly	5	0.97
Definition of characteristic polynomial of a linear transformation
math.la.c.lintrans.eig.arb	math.la.d.lintrans.minpoly	5	0.96
Definition of minimal polynomial of a linear transformation
math.la.c.lintrans.eig.arb	math.la.d.lintrans.cayleyhamilton	5	0.95
The Cayley-Hamilton theorem for a linear transformation
math.la.c.lintrans.eig.arb	math.la.d.lintrans.minpoly.exists	5	0.94
The minimal polynomial of a linear transformation exists and is unique.
math.la.c.lintrans.eig.arb	math.la.d.lintrans.polynomial.apply	5	0.93
Definition of applying a polynomial to a linear transformation
math.la.c.lintrans.eig.arb	math.la.t.lintrans.eig.exists	5	0.92
A linear transformation on a finite dimentional nontrivial vector space has at least one eigenvalue.
math.la.c.lintrans.eig.arb	math.la.d.lintrans.eigsp	5	0.91
Definition of eigenspace of a linear transformation
math.la.c.lintrans.eig.arb	math.la.d.lintrans.eigsp.subspace	5	0.90
The eigenspace of a linear transformation is a nontrivial subspace.
math.la.c.lintrans.eig.arb	math.la.d.lintrans.invariantsubspace	5	0.89
Definition of invariant subspace of a linear transformation.
math.la.c.lintrans.eig.arb	math.la.d.lintrans.invariantsubspace.block	5	0.88
If a space is the direct sum of invariant subspaces, then the linear transformation has a block diagonal representation.
math.la.c.lintrans.arb	math.la.t.lintrans.diagonalizable.basis	4	0.85
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
math.la.c.lintrans.arb	math.la.c.lintrans.subspace.arb	4	0.84
Subspaces associated to a linear transformation
math.la.c.lintrans.subspace.arb	math.la.t.lintrans.range.vsp	5	0.99
The range/image of a linear transformation is a subspace.
math.la.c.lintrans.subspace.arb	math.la.t.lintrans.inv.vsp	5	0.98
The inverse image of a subspace under a linear transformation is a subspace.
math.la.c.lintrans.subspace.arb	math.la.d.lintrans.kernel.arb	5	0.97
Definition of kernel/null space of linear transformation, arbitrary vector space
math.la.c.lintrans.subspace.arb	math.la.t.lintrans.kernel.arb	5	0.96
The kernel/null space of a linear transformation is a subspace, arbitrary vector space
math.la.c.lintrans.subspace.arb	math.la.d.lintrans.kernel.generalized.arb	5	0.95
Definition of generalized kernel/null space of linear transformation, arbitrary vector space
math.la.c.lintrans.subspace.arb	math.la.d.lintrans.generalized.sum	5	0.94
Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.
math.la.c.lintrans.subspace.arb	math.la.t.lintrans.power.kernel	5	0.93
The kernels of powers of a linear transformation form an ascending chain, with proper containment up to a particular power and then equality for all subsequent powers.
math.la.c.lintrans.subspace.arb	math.la.t.lintrans.power.range	5	0.92
The range spaces of powers of a linear transformation form a descending chain, with proper containment up to a particular power and then equality for all subsequent powers.
math.la.c.lintrans.arb	math.la.t.lintrans.ranknullity	4	0.83
The rank plus the nullity of a linear transformation equals the dimension of the domain.
math.la.c.lintrans.arb	math.la.t.lintrans.lindep	4	0.82
The image of a linearly dependent set under a linear transformation is linearly dependent.
math.la.c.lintrans.arb	math.la.t.lintrans.onto.rank	4	0.81
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
math.la.c.lintrans.arb	math.la.t.lintrans.invertible.arb	4	0.80
A linear transformation is invertible if and only if it is injective and surjective, arbitrary vector space
math.la.c.lintrans.arb	math.la.d.lintrans.mat.repn.arb	4	0.79
Definition of matrix representation of a linear transformation with respect to bases of the spaces, arbitrary vector space
math.la.c.lintrans.arb	math.la.t.lintrans.mat.repn.arb	4	0.78
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces, arbitrary vector space.
math.la.c.lintrans.arb	math.la.d.lintrans.mat.repn.self.arb	4	0.77
Definition of matrix representation of a linear transformation from a vector space to itself, with respect to basis of the space, arbitrary vector space
math.la.c.lintrans.arb	math.la.t.lintrans.mat_repn.scalar	4	0.76
The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
math.la.c.lintrans.arb	math.la.t.lintrans.mat_repn.sum	4	0.75
The matrix representation of a sum of linear transformations is the sum of the matrices.
math.la.c.lintrans.arb	math.la.t.lintrans.mat_repn.composition	4	0.74
The matrix representation of a composition of linear transformations is the product of the matrices.
math.la.c.lintrans.arb	math.la.t.lintrans.mat_repn.inv	4	0.73
The matrix representation of the inverse of linear transformations is the inverse of the matricix.
math.la.c.lintrans.arb	math.la.t.lintrans.mat_repn.eig	4	0.72
A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.
math.la.c.lintrans.arb	math.la.t.lintrans.mat_repn.triangular	4	0.71
A linear transformation has a representation as an upper triangular matrix.
math.la.c.lintrans.arb	math.la.c.lintrans.equiv	4	0.70
Equivalence theorems for injective transformations
math.la.c.lintrans.equiv	math.la.d.lintrans.injective.arb	5	0.99
Definition of one-to-one/injective linear transformation, arbitrary vector space
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.inv	5	0.98
Equivalence theorem for injective linear transformations: The inverse of T is a linear transformation on its range.
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.nullspace	5	0.97
Equivalence theorem for injective linear transformations: The null space of T is 0.
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.nullity	5	0.96
Equivalence theorem for injective linear transformations: The nullity of T is 0.
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.kernel	5	0.95
Equivalence theorem for injective linear transformations: The kernel of T is 0.
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.z	5	0.94
Equivalence theorem for injective linear transformations: T(x)=0 only for x=0.
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.rank	5	0.93
Equivalence theorem for injective linear transformations: The rank of T is n.
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.rank.col	5	0.92
Equivalence theorem for injective linear transformations: The rank of T is equals the number of columns in any matrix representation..
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.basis	5	0.91
Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.
math.la.c.lintrans.equiv	math.la.t.lintrans.equiv.linindep	5	0.90
Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.
math.la.c.vsp.abs	math.la.c.vec.orthog_proj.arb	3	0.89
Orthogonality and projection
math.la.c.vec.orthog_proj.arb	math.la.c.innerproduct.arb	4	0.99
Inner products
math.la.c.innerproduct.arb	math.la.d.vec.innerproduct.arb	5	0.99
Definition of inner product, arbitrary setting
math.la.c.innerproduct.arb	math.la.d.vsp.innerproduct.arb	5	0.98
Definition of inner product space, arbitrary setting
math.la.c.vec.orthog_proj.arb	math.la.c.vec.norm.coord	4	0.98
Norm and length
math.la.c.vec.norm.coord	math.la.d.vec.norm.arb	5	0.99
Definition of length/norm of a vector, arbitrary setting
math.la.c.vec.norm.coord	math.la.d.vec.distance.arb	5	0.98
Definition of distance between vectors, arbitrary setting
math.la.c.vec.norm.coord	math.la.t.vec.cauchyschwarz.arb	5	0.97
The Cauchy-Schwarz inequality, arbitrary setting
math.la.c.vec.norm.coord	math.la.t.vec.triangle.arb	5	0.96
The triangle inequality, arbitrary setting
math.la.c.vec.orthog_proj.arb	math.la.c.vec.orthogonal.arb	4	0.97
Orthogonality
math.la.c.vec.orthogonal.arb	math.la.d.vec.orthogonal.arb	5	0.99
Definition of orthogonal vectors, arbitrary setting
math.la.c.vec.orthogonal.arb	math.la.d.vec.parallel.arb	5	0.98
Definition of parallel vectors, arbitrary setting
math.la.c.vec.orthogonal.arb	math.la.d.gramschmidt.arb	5	0.97
Definition of Gram-Schmidt process, arbitrary setting
math.la.c.vec.orthogonal.arb	math.la.d.vsp.orthogonal.arb	5	0.96
Definition of orthogonal subspaces, arbitrary setting
math.la.c.vec.orthog_proj.arb	math.la.c.vec.proj.arb	4	0.96
Projection
math.la.c.vsp	math.la.c.vsp.example	2	0.97
Examples of vector spaces
math.la.c.vsp.example	math.la.e.vsp.z	3	0.99
The set containing only 0 is a vector space.
math.la.c.vsp.example	math.la.e.vsp.rn	3	0.98
R^n is a vector space.
math.la.c.vsp.example	math.la.e.vsp.cn	3	0.97
C^n is a vector space.
math.la.c.vsp.example	math.la.e.vsp.fn	3	0.96
F^n is a vector space.
math.la.c.vsp.example	math.la.e.vsp.function	3	0.95
The set of all functions on a set is a vector space.
math.la.c.vsp.example	math.la.e.vsp.linsys.homog	3	0.94
The solutions to a homogeneous system of linear equations is a vector space.
math.la.c.vsp.example	math.la.e.vsp.de.homog	3	0.93
The solutions to a homogeneous linear differential equation is a vector space.
math.la.c.vsp.example	math.la.e.vsp.polynomial	3	0.92
The set of all polynomials is a vector space.
math.la.c.vsp.example	math.la.e.vsp.polynomial.leq_n	3	0.91
The set of all polynomials of degree at most n is a vector space.
math.la.c.vsp.example	math.la.e.vsp.mat.m_by_n	3	0.90
The set of m by n matrices is a vector space.
math.la.c.vsp.example	math.la.e.vsp.sequence	3	0.89
The set of all sequences is a vector space.
math.la.c.vsp.example	math.la.e.vsp.function	3	0.88
The set of all functions on a set is a vector space.
math.la.c.vsp.example	math.la.e.vsp.crazy	3	0.87
The crazy vector space is a vector space.
math.la.c.vsp.example	math.la.e.vsp.row	3	0.86
The row space of a matrix is a vector space
math.la.c.vsp.example	math.la.e.vsp.col	3	0.85
The column space of a matrix is a vector space
math.la.c.vsp.example	math.la.t.mat.null_space.rncn	3	0.84
The null space of a matrix is a subspace of R^n (or C^n).
math.la.c.vsp.example	math.la.t.mat.null_space.left.rncn	3	0.83
The left null space of a matrix is a subspace of R^m (or C^m).
math.la.c.vsp.example	math.la.t.lintrans.vsp	3	0.82
The set of linear transformations between two vector spaces is a vector space.
math.la	math.la.c.applications	1	0.95
Applications
math.la.c.applications	math.la.a.differentialequation	2	0.99
Applications to differential equations
math.la.c.applications	math.la.a.markovchain	2	0.98
Applications to Markov chains
math.la.c.applications	math.la.a.bandmatrix	2	0.97
Applications of band matrices
math.la.c.applications	math.la.a.errorcorrectingcode	2	0.96
Applications to error-correcting code
math.la.c.applications	math.la.a.inputoutput	2	0.95
Application Leontief input-output analysis
math.la.c.applications	math.la.a.voting	2	0.94
Applications to voting and social choice
math.la.c.applications	math.la.a.cubicspline	2	0.93
Applications to cubic spline
math.la	math.la.c.other	1	0.94
Other
math.la.c.other	math.la.d.crossproduct	2	0.99
Definition of cross product
math.la.c.other	math.la.d.quadraticform	2	0.98
Definition of quadratic form, orthonormal diagonalization, principal axes