In this section we specialize to systems of linear equations where every equation has a zero as its constant term. Along the way, we will begin to express more and more ideas in the language of matrices and begin a move away from writing out whole systems of equations. The ideas initiated in this section will carry through the remainder of the course.

## Solutions of Homogeneous Systems

As usual, we begin with a definition.

Definition HS (Homogeneous System) A system of linear equations, $\linearsystem{A}{\vect{b}}$ is homogeneous if the vector of constants is the zero vector, in other words, $\vect{b}=\zerovector$.

Example AHSAC: Archetype C as a homogeneous system.

As you might have discovered by studying Example AHSAC, setting each variable to zero will always be a solution of a homogeneous system. This is the substance of the following theorem.

Theorem HSC (Homogeneous Systems are Consistent) Suppose that a system of linear equations is homogeneous. Then the system is consistent.

Since this solution is so obvious, we now define it as the trivial solution.

Definition TSHSE (Trivial Solution to Homogeneous Systems of Equations) Suppose a homogeneous system of linear equations has $n$ variables. The solution $x_1=0$, $x_2=0$,..., $x_n=0$ (i.e. $\vect{x}=\zerovector$) is called the trivial solution.

Here are three typical examples, which we will reference throughout this section. Work through the row operations as we bring each to reduced row-echelon form. Also notice what is similar in each example, and what differs.

Example HUSAB: Homogeneous, unique solution, Archetype B.

Example HISAA: Homogeneous, infinite solutions, Archetype A.

Example HISAD: Homogeneous, infinite solutions, Archetype D.

After working through these examples, you might perform the same computations for the slightly larger example, Archetype J.

Notice that when we do row operations on the augmented matrix of a homogeneous system of linear equations the last column of the matrix is all zeros. Any one of the three allowable row operations will convert zeros to zeros and thus, the final column of the matrix in reduced row-echelon form will also be all zeros. So in this case, we may be as likely to reference only the coefficient matrix and presume that we remember that the final column begins with zeros, and after any number of row operations is still zero.

Example HISAD suggests the following theorem.

Theorem HMVEI (Homogeneous, More Variables than Equations, Infinite solutions) Suppose that a homogeneous system of linear equations has $m$ equations and $n$ variables with $n>m$. Then the system has infinitely many solutions.

Example HUSAB and Example HISAA are concerned with homogeneous systems where $n=m$ and expose a fundamental distinction between the two examples. One has a unique solution, while the other has infinitely many. These are exactly the only two possibilities for a homogeneous system and illustrate that each is possible (unlike the case when $n>m$ where Theorem HMVEI tells us that there is only one possibility for a homogeneous system).

## Null Space of a Matrix

The set of solutions to a homogeneous system (which by Theorem HSC is never empty) is of enough interest to warrant its own name. However, we define it as a property of the coefficient matrix, not as a property of some system of equations.

Definition NSM (Null Space of a Matrix) The null space of a matrix $A$, denoted $\nsp{A}$, is the set of all the vectors that are solutions to the homogeneous system $\homosystem{A}$.

In the Archetypes (appendix A) each example that is a system of equations also has a corresponding homogeneous system of equations listed, and several sample solutions are given. These solutions will be elements of the null space of the coefficient matrix. We'll look at one example.

Example NSEAI: Null space elements of Archetype I.

Here are two (prototypical) examples of the computation of the null space of a matrix.

Example CNS1: Computing a null space, \#1.

Example CNS2: Computing a null space, \#2.