In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

## Invertible Linear Transformations

One preliminary definition, and then we will have our main definition for this section.

Definition IDLT (Identity Linear Transformation) The identity linear transformation on the vector space $W$ is defined as \begin{equation*} \ltdefn{I_W}{W}{W},       \lt{I_W}{\vect{w}}=\vect{w} \end{equation*}

Informally, $I_W$ is the "do-nothing" function. You should check that $I_W$ is really a linear transformation, as claimed, and then compute its kernel and range to see that it is both injective and surjective. All of these facts should be straightforward to verify (exercise IVLT.T05). With this in hand we can make our main definition.

Definition IVLT (Invertible Linear Transformations) Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. If there is a function $\ltdefn{S}{V}{U}$ such that

\begin{align*} \compose{S}{T}&=I_U & \compose{T}{S}&=I_V \end{align*}

then $T$ is invertible. In this case, we call $S$ the inverse of $T$ and write $S=\ltinverse{T}$.

Informally, a linear transformation $T$ is invertible if there is a companion linear transformation, $S$, which "undoes" the action of $T$. When the two linear transformations are applied consecutively (composition), in either order, the result is to have no real effect. It is entirely analogous to squaring a positive number and then taking its (positive) square root.

Here is an example of a linear transformation that is invertible. As usual at the beginning of a section, do not be concerned with where $S$ came from, just understand how it illustrates Definition IVLT.

Example AIVLT: An invertible linear transformation.

It can be as instructive to study a linear transformation that is not invertible.

Example ANILT: A non-invertible linear transformation.

In Example ANILT you may have noticed that $T$ is not surjective, since the matrix $A$ was not in the range of $T$. And $T$ is not injective since there are two different input column vectors that $T$ sends to the matrix $B$. Linear transformations $T$ that are not surjective lead to putative inverse functions $S$ that are undefined on inputs outside of the range of $T$. Linear transformations $T$ that are not injective lead to putative inverse functions $S$ that are multiply-defined on each of their inputs. We will formalize these ideas in Theorem ILTIS.

But first notice in Definition IVLT that we only require the inverse (when it exists) to be a function. When it does exist, it too is a linear transformation.

Theorem ILTLT (Inverse of a Linear Transformation is a Linear Transformation) Suppose that $\ltdefn{T}{U}{V}$ is an invertible linear transformation. Then the function $\ltdefn{\ltinverse{T}}{V}{U}$ is a linear transformation.

So $\ltinverse{T}$ fulfills the requirements of Definition LT and is therefore a linear transformation. So when $T$ has an inverse, $\ltinverse{T}$ is also a linear transformation. Additionally, $\ltinverse{T}$ is invertible and its inverse is what you might expect.

Theorem IILT (Inverse of an Invertible Linear Transformation) Suppose that $\ltdefn{T}{U}{V}$ is an invertible linear transformation. Then $\ltinverse{T}$ is an invertible linear transformation and $\ltinverse{\left(\ltinverse{T}\right)}=T$.

## Invertibility

We now know what an inverse linear transformation is, but just which linear transformations have inverses? Here is a theorem we have been preparing for all chapter long.

Theorem ILTIS (Invertible Linear Transformations are Injective and Surjective) Suppose $\ltdefn{T}{U}{V}$ is a linear transformation. Then $T$ is invertible if and only if $T$ is injective and surjective.

When a linear transformation is both injective and surjective, the pre-image of any element of the codomain is a set of size one (a "singleton"). This fact allowed us to construct the inverse linear transformation in one half of the proof of Theorem ILTIS (see technique C). We can follow this approach to construct the inverse of a specific linear transformation, as the next example shows.

Example CIVLT: Computing the Inverse of a Linear Transformations.

We will make frequent use of the characterization of invertible linear transformations provided by Theorem ILTIS. The next theorem is a good example of this, and we will use it often, too.

Theorem CIVLT (Composition of Invertible Linear Transformations) Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are invertible linear transformations. Then the composition, $\ltdefn{\left(\compose{S}{T}\right)}{U}{W}$ is an invertible linear transformation.

When a composition is invertible, the inverse is easy to construct.

Theorem ICLT (Inverse of a Composition of Linear Transformations) Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are invertible linear transformations. Then $\compose{S}{T}$ is invertible and $\ltinverse{\left(\compose{S}{T}\right)}=\compose{\ltinverse{T}}{\ltinverse{S}}$.

Notice that this theorem not only establishes what the inverse of $\compose{S}{T}$ is, it also duplicates the conclusion of Theorem CIVLT and also establishes the invertibility of $\compose{S}{T}$. But somehow, the proof of Theorem CIVLT is nicer way to get this property.

Does Theorem ICLT remind you of the flavor of any theorem we have seen about matrices? (Hint: Think about getting dressed.) Hmmmm.

## Structure and Isomorphism

A vector space is defined (Definition VS) as a set of objects ("vectors") endowed with a definition of vector addition ($+$) and a definition of scalar multiplication (written with juxtaposition). Many of our definitions about vector spaces involve linear combinations (Definition LC), such as the span of a set (Definition SS) and linear independence (Definition LI). Other definitions are built up from these ideas, such as bases (Definition B) and dimension (Definition D). The defining properties of a linear transformation require that a function "respect" the operations of the two vector spaces that are the domain and the codomain (Definition LT). Finally, an invertible linear transformation is one that can be "undone" --- it has a companion that reverses its effect. In this subsection we are going to begin to roll all these ideas into one.

A vector space has "structure" derived from definitions of the two operations and the requirement that these operations interact in ways that satisfy the ten properties of Definition VS. When two different vector spaces have an invertible linear transformation defined between them, then we can translate questions about linear combinations (spans, linear independence, bases, dimension) from the first vector space to the second. The answers obtained in the second vector space can then be translated back, via the inverse linear transformation, and interpreted in the setting of the first vector space. We say that these invertible linear transformations "preserve structure." And we say that the two vector spaces are "structurally the same." The precise term is "isomorphic," from Greek meaning "of the same form." Let's begin to try to understand this important concept.

Definition IVS (Isomorphic Vector Spaces) Two vector spaces $U$ and $V$ are isomorphic if there exists an invertible linear transformation $T$ with domain $U$ and codomain $V$, $\ltdefn{T}{U}{V}$. In this case, we write $U\isomorphic V$, and the linear transformation $T$ is known as an isomorphism between $U$ and $V$.

A few comments on this definition. First, be careful with your language (technique L). Two vector spaces are isomorphic, or not. It is a yes/no situation and the term only applies to a pair of vector spaces. Any invertible linear transformation can be called an isomorphism, it is a term that applies to functions. Second, a given pair of vector spaces there might be several different isomorphisms between the two vector spaces. But it only takes the existence of one to call the pair isomorphic. Third, $U$ isomorphic to $V$, or $V$ isomorphic to $U$? Doesn't matter, since the inverse linear transformation will provide the needed isomorphism in the "opposite" direction. Being "isomorphic to" is an equivalence relation on the set of all vector spaces (see Theorem SER for a reminder about equivalence relations).

Example IVSAV: Isomorphic vector spaces, Archetype V.

Checking the dimensions of two vector spaces can be a quick way to establish that they are not isomorphic. Here's the theorem.

Theorem IVSED (Isomorphic Vector Spaces have Equal Dimension) Suppose $U$ and $V$ are isomorphic vector spaces. Then $\dimension{U}=\dimension{V}$.

The contrapositive of Theorem IVSED says that if $U$ and $V$ have different dimensions, then they are not isomorphic. Dimension is the simplest "structural" characteristic that will allow you to distinguish non-isomorphic vector spaces. For example $P_6$ is not isomorphic to $M_{34}$ since their dimensions (7 and 12, respectively) are not equal. With tools developed in Section VR:Vector Representations we will be able to establish that the converse of Theorem IVSED is true. Think about that one for a moment.

## Rank and Nullity of a Linear Transformation

Just as a matrix has a rank and a nullity, so too do linear transformations. And just like the rank and nullity of a matrix are related (they sum to the number of columns, Theorem RPNC) the rank and nullity of a linear transformation are related. Here are the definitions and theorems, see the Archetypes (appendix A) for loads of examples.

Definition ROLT (Rank Of a Linear Transformation) Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the rank of $T$, $\rank{T}$, is the dimension of the range of $T$, \begin{equation*} \rank{T}=\dimension{\rng{T}} \end{equation*}

Definition NOLT (Nullity Of a Linear Transformation) Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the nullity of $T$, $\nullity{T}$, is the dimension of the kernel of $T$, \begin{equation*} \nullity{T}=\dimension{\krn{T}} \end{equation*}

Here are two quick theorems.

Theorem ROSLT (Rank Of a Surjective Linear Transformation) Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the rank of $T$ is the dimension of $V$, $\rank{T}=\dimension{V}$, if and only if $T$ is surjective.

Theorem NOILT (Nullity Of an Injective Linear Transformation) Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the nullity of $T$ is zero, $\nullity{T}=0$, if and only if $T$ is injective.

Just as injectivity and surjectivity come together in invertible linear transformations, there is a clear relationship between rank and nullity of a linear transformation. If one is big, the other is small.

Theorem RPNDD (Rank Plus Nullity is Domain Dimension) Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then \begin{equation*} \rank{T}+\nullity{T}=\dimension{U} \end{equation*}

Theorem RPNC said that the rank and nullity of a matrix sum to the number of columns of the matrix. This result is now an easy consequence of Theorem RPNDD when we consider the linear transformation $\ltdefn{T}{\complex{n}}{\complex{m}}$ defined with the $m\times n$ matrix $A$ by $\lt{T}{\vect{x}}=A\vect{x}$. The range and kernel of $T$ are identical to the column space and null space of the matrix $A$ (exercise ILT.T20, exercise SLT.T20), so the rank and nullity of the matrix $A$ are identical to the rank and nullity of the linear transformation $T$. The dimension of the domain of $T$ is the dimension of $\complex{n}$, exactly the number of columns for the matrix $A$.

This theorem can be especially useful in determining basic properties of linear transformations. For example, suppose that $\ltdefn{T}{\complex{6}}{\complex{6}}$ is a linear transformation and you are able to quickly establish that the kernel is trivial. Then $\nullity{T}=0$. First this means that $T$ is injective by Theorem NOILT. Also, Theorem RPNDD becomes \begin{equation*} 6=\dimension{\complex{6}}=\rank{T}+\nullity{T}=\rank{T}+0=\rank{T} \end{equation*} So the rank of $T$ is equal to the rank of the codomain, and by Theorem ROSLT we know $T$ is surjective. Finally, we know $T$ is invertible by Theorem ILTIS. So from the determination that the kernel is trivial, and consideration of various dimensions, the theorems of this section allow us to conclude the existence of an inverse linear transformation for $T$.

Similarly, Theorem RPNDD can be used to provide alternative proofs for Theorem ILTD, Theorem SLTD and Theorem IVSED. It would be an interesting exercise to construct these proofs.

It would be instructive to study the archetypes that are linear transformations and see how many of their properties can be deduced just from considering only the dimensions of the domain and codomain. Then add in just knowledge of either the nullity or rank, and so how much more you can learn about the linear transformation. The table preceding all of the archetypes (appendix A) could be a good place to start this analysis.

## Systems of Linear Equations and Linear Transformations

This subsection does not really belong in this section, or any other section, for that matter. It is just the right time to have a discussion about the connections between the central topic of linear algebra, linear transformations, and our motivating topic from Chapter SLE:Systems of Linear Equations, systems of linear equations. We will discuss several theorems we have seen already, but we will also make some forward-looking statements that will be justified in Chapter R:Representations.

Archetype D and Archetype E are ideal examples to illustrate connections with linear transformations. Both have the same coefficient matrix, \begin{equation*} D= \begin{bmatrix} 2 & 1 & 7 & -7\\ -3 & 4 & -5 & -6\\ 1 & 1 & 4 & -5 \end{bmatrix} \end{equation*} To apply the theory of linear transformations to these two archetypes, employ matrix multiplication (Definition MM) and define the linear transformation, \begin{equation*} \ltdefn{T}{\complex{4}}{\complex{3}},    \lt{T}{\vect{x}}=D\vect{x} =x_1\colvector{2\\-3\\1}+ x_2\colvector{1\\4\\1}+ x_3\colvector{7\\-5\\4}+ x_4\colvector{-7\\-6\\-5} \end{equation*} Theorem MBLT tells us that $T$ is indeed a linear transformation. Archetype D asks for solutions to $\linearsystem{D}{\vect{b}}$, where $\vect{b}=\colvector{8\\-12\\-4}$. In the language of linear transformations this is equivalent to asking for $\preimage{T}{\vect{b}}$. In the language of vectors and matrices it asks for a linear combination of the four columns of $D$ that will equal $\vect{b}$. One solution listed is $\vect{w}=\colvector{7\\8\\1\\3}$. With a non-empty preimage, Theorem KPI tells us that the complete solution set of the linear system is the preimage of $\vect{b}$, \begin{equation*} \vect{w}+\krn{T}=\setparts{\vect{w}+\vect{z}}{\vect{z}\in\krn{T}} \end{equation*} The kernel of the linear transformation $T$ is exactly the null space of the matrix $D$ (see exercise ILT.T20), so this approach to the solution set should be reminiscent of Theorem PSPHS. The kernel of the linear transformation is the preimage of the zero vector, exactly equal to the solution set of the homogeneous system $\homosystem{D}$. Since $D$ has a null space of dimension two, every preimage (and in particular the preimage of $\vect{b}$) is as "big" as a subspace of dimension two (but is not a subspace).

Archetype E is identical to Archetype D but with a different vector of constants, $\vect{d}=\colvector{2\\3\\2}$. We can use the same linear transformation $T$ to discuss this system of equations since the coefficient matrix is identical. Now the set of solutions to $\linearsystem{D}{\vect{d}}$ is the pre-image of $\vect{d}$, $\preimage{T}{\vect{d}}$. However, the vector $\vect{d}$ is not in the range of the linear transformation (nor is it in the column space of the matrix, since these two sets are equal by exercise SLT.T20). So the empty pre-image is equivalent to the inconsistency of the linear system.

These two archetypes each have three equations in four variables, so either the resulting linear systems are inconsistent, or they are consistent and application of Theorem CMVEI tells us that the system has infinitely many solutions. Considering these same parameters for the linear transformation, the dimension of the domain, $\complex{4}$, is four, while the codomain, $\complex{3}$, has dimension three. Then

\begin{align*} \nullity{T}&=\dimension{\complex{4}}-\rank{T}&&\knowl{RPNDDtheorem.knowl}{\text{Theorem RPNDD}}\\ &=4-\dimension{\rng{T}}&&\knowl{ROLTdefinition.knowl}{\text{Definition ROLT}}\\ &\geq 4-3&&\text{$\rng{T}$ subspace of $\complex{3}$}\\ &=1 \end{align*}

So the kernel of $T$ is nontrivial simply by considering the dimensions of the domain (number of variables) and the codomain (number of equations). Pre-images of elements of the codomain that are not in the range of $T$ are empty (inconsistent systems). For elements of the codomain that are in the range of $T$ (consistent systems), Theorem KPI tells us that the pre-images are built from the kernel, and with a non-trivial kernel, these pre-images are infinite (infinitely many solutions).

When do systems of equations have unique solutions? Consider the system of linear equations $\linearsystem{C}{\vect{f}}$ and the linear transformation $\lt{S}{\vect{x}}=C\vect{x}$. If $S$ has a trivial kernel, then pre-images will either be empty or be finite sets with single elements. Correspondingly, the coefficient matrix $C$ will have a trivial null space and solution sets will either be empty (inconsistent) or contain a single solution (unique solution). Should the matrix be square and have a trivial null space then we recognize the matrix as being nonsingular. A square matrix means that the corresponding linear transformation, $T$, has equal-sized domain and codomain. With a nullity of zero, $T$ is injective, and also Theorem RPNDD tells us that rank of $T$ is equal to the dimension of the domain, which in turn is equal to the dimension of the codomain. In other words, $T$ is surjective. Injective and surjective, and Theorem ILTIS tells us that $T$ is invertible. Just as we can use the inverse of the coefficient matrix to find the unique solution of any linear system with a nonsingular coefficient matrix (Theorem SNCM), we can use the inverse of the linear transformation to construct the unique element of any pre-image (proof of Theorem ILTIS).

The executive summary of this discussion is that to every coefficient matrix of a system of linear equations we can associate a natural linear transformation. Solution sets for systems with this coefficient matrix are preimages of elements of the codomain of the linear transformation. For every theorem about systems of linear equations there is an analogue about linear transformations. The theory of linear transformations provides all the tools to recreate the theory of solutions to linear systems of equations.

We will continue this adventure in Chapter R:Representations.