Section IVLT  Invertible Linear Transformations

From A First Course in Linear Algebra
Version 1.08
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/

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.

Subsection IVLT: 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

IW : WW,IW w = w

Informally, IW is the “do-nothing” function. You should check that IW 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 T : UV is a linear transformation. If there is a function S : V U such that

S T = IU T S = IV

then T is invertible. In this case, we call S the inverse of T and write S = T1.

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
Archetype V is the linear transformation

T : P3M22,T a + bx + cx2 + dx3 = a + ba 2c d b d

Define the function S : M22P3 defined by

S ab cd = (acd)+(c+d)x+1 2(abcd)x2+cx3

Then

T S ab cd = T S ab cd = T (a c d) + (c + d)x + 1 2(a b c d)x2 + cx3 = (a c d) + (c + d)(a c d) 2(1 2(a b c d)) c (c + d) c = ab cd = IM22 ab cd  And S T a + bx + cx2 + dx3 = S T a + bx + cx2 + dx3 = S a + ba 2c d b d = ((a + b) d (b d)) + (d + (b d))x + 1 2((a + b) (a 2c) d (b d)) x2 + (d)x3 = a + bx + cx2 + dx3 = IP3 a + bx + cx2 + dx3

For now, understand why these computations show that T is invertible, and that S = T1. Maybe even be amazed by how S works so perfectly in concert with T! We will see later just how to arrive at the correct form of S (when it is possible).

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

Example ANILT
A non-invertible linear transformation
Consider the linear transformation T : 3M 22 defined by

T a b c = a b 2a + 2b + c 3a + b + c2a 6b 2c

Suppose we were to search for an inverse function S : M223.

First verify that the 2 × 2 matrix A = 53 82 is not in the range of T. This will amount to finding an input to T, a b c , such that

a b = 5 2a + 2b + c = 3 3a + b + c = 8 2a 6b 2c = 2

As this system of equations is inconsistent, there is no input column vector, and AT. How should we define S A? Note that

T S A = T S A = IM22 A = A

So any definition we would provide for S A must then be a column vector that T sends to A and we would have A T, contrary to the definition of T. This is enough to see that there is no function S that will allow us to conclude that T is invertible, since we cannot provide a consistent definition for S A if we assume T is invertible.

Even though we now know that T is not invertible, let’s not leave this example just yet. Check that

T 1 2 4 = 32 52 = B T 0 3 8 = 32 52 = B

How would we define S B?

S B = S T 1 2 4 = S T 1 2 4 = I3 1 2 4 = 1 2 4  or S B = S T 0 3 8 = S T 0 3 8 = I3 0 3 8 = 0 3 8

Which definition should we provide for S B? Both are necessary. But then S is not a function. So we have a second reason to know that there is no function S that will allow us to conclude that T is invertible. It happens that there are infinitely many column vectors that S would have to take to B. Construct the kernel of T,

KT = 1 1 4

Now choose either of the two inputs used above for T and add to it a scalar multiple of the basis vector for the kernel of T. For example,

x = 1 2 4 +(2) 1 1 4 = 3 0 4

then verify that T x = B. Practice creating a few more inputs for T that would be sent to B, and see why it is hopeless to think that we could ever provide a reasonable definition for S B! There is a “whole subspace’s worth” of values that S B would have to take on.

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 T : UV is an invertible linear transformation. Then the function T1: V U is a linear transformation.

Proof   We work through verifying Definition LT for T1, using the fact that T is a linear transformation to obtain the second equality in each half of the proof. To this end, suppose x,y V and α .

T1 x + y = T1 T T1 x + T T1 y  Definition IVLT = T1 T T1 x + T1 y  Definition LT = T1 x + T1 y  Definition IVLT  Now check the second defining property of a linear transformation for T1, T1 αx = T1 αT T1 x  Definition IVLT = T1 T αT1 x  Definition LT = αT1 x  Definition IVLT

So T1 fulfills the requirements of Definition LT and is therefore a linear transformation. So when T has an inverse, T1 is also a linear transformation. Additionally, T1 is invertible and its inverse is what you might expect.

Theorem IILT
Inverse of an Invertible Linear Transformation
Suppose that T : UV is an invertible linear transformation. Then T1 is an invertible linear transformation and T1 1 = T.

Proof   Because T is invertible, Definition IVLT tells us there is a function T1: V U such that

T1 T = I U T T1 = I V

Additionally, Theorem ILTLT tells us that T1 is more than just a function, it is a linear transformation. Now view these two statements as properties of the linear transformation T1. In light of Definition IVLT, they together say that T1 is invertible (let T play the role of S in the statement of the definition). Furthermore, the inverse of T1 is then T, i.e.  T1 1 = T.

Subsection IV: 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 T : UV is a linear transformation. Then T is invertible if and only if T is injective and surjective.

Proof   ( ) Since T is presumed invertible, we can employ its inverse, T1 (Definition IVLT). To see that T is injective, suppose x,y U and assume that T x = T y,

x = IU x  Definition IDLT = T1 T x  Definition IVLT = T1 T x  Definition LTC = T1 T y  Definition ILT = T1 T y  Definition LTC = IU y  Definition IVLT = y  Definition IDLT   So by Definition ILT T is injective. To check that T is surjective, suppose v V . Then T1 v is a vector in U. Compute  T T1 v = T T1 v  Definition LTC = IV v  Definition IVLT = v  Definition IDLT

So there is an element from U, when used as an input to T (namely T1 v) that produces the desired output, v, and hence T is surjective by Definition SLT.

( ) Now assume that T is both injective and surjective. We will build a function S : V U that will establish that T is invertible. To this end, choose any v V . Since T is surjective, Theorem RSLT says T = V , so we have v T. Theorem RPI says that the pre-image of v, T1 v, is nonempty. So we can choose a vector from the pre-image of v, say u. In other words, there exists u T1 v.

Since T1 v is non-empty, Theorem KPI then says that

T1 v = u + zz KT

However, because T is injective, by Theorem KILT the kernel is trivial, KT = 0. So the pre-image is a set with just one element, T1 v = u. Now we can define S by S v = u. This is the key to this half of this proof. Normally the preimage of a vector from the codomain might be an empty set, or an infinite set. But surjectivity requires that the preimage not be empty, and then injectivity limits the preimage to a singleton. Since our choice of v was arbitrary, we know that every pre-image for T is a set with a single element. This allows us to construct S as a function. Now that it is defined, verifying that it is the inverse of T will be easy. Here we go.

Choose u U. Define v = T u. Then T1 v = u, so that S v = u and,

S T u = S T u = S v = u = IU u

and since our choice of u was arbitrary we have function equality, S T = IU.

Now choose v V . Define u to be the single vector in the set T1 v, in other words, u = S v. Then T u = v, so

T S v = T S v = T u = v = IV v

and since our choice of v was arbitary we have function equality, T S = IV .

We will make frequent use of this characterization of invertible linear transformations. 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 T : UV and S : V W are invertible linear transformations. Then the composition, S T : UW is an invertible linear transformation.

Proof   Since S and T are both linear transformations, S T is also a linear transformation by Theorem CLTLT. Since S and T are both invertible, Theorem ILTIS says that S and T are both injective and surjective. Then Theorem CILTI says S T is injective, and Theorem CSLTS says S T is surjective. Now apply the “other half” of Theorem ILTIS and conclude that S T is invertible.

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

Theorem ICLT
Inverse of a Composition of Linear Transformations
Suppose that T : UV and S : V W are invertible linear transformations. Then S T is invertible and S T1 = T1 S1.

Proof   Compute, for all w W

S T T1 S1 w = S T T1 S1 w = S IV S1 w  Definition IVLT = S S1 w  Definition IDLT = w  Definition IVLT = IW w  Definition IDLT   so S T T1 S1 = I W  and also T1 S1 S T u = T1 S1 S T u = T1 I V T u  Definition IVLT = T1 T u  Definition IDLT = u  Definition IVLT = IU u  Definition IDLT

so T1 S1 S T = I U. By Definition IVLT, S T is invertible and S T1 = T1 S1.

Notice that this theorem not only establishes what the inverse of S T is, it also duplicates the conclusion of Theorem CIVLT and also establishes the invertibility of 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.

Subsection SI: 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 , T : UV . In this case, we write UV , 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
Archetype V is a linear transformation from P3 to M22,

T : P3M22,T a + bx + cx2 + dx3 = a + ba 2c d b d

Since it is injective and surjective, Theorem ILTIS tells us that it is an invertible linear transformation. By Definition IVS we say P3 and M22 are isomorphic.

At a basic level, the term “isomorphic” is nothing more than a codeword for the presence of an invertible linear transformation. However, it is also a description of a powerful idea, and this power only becomes apparent in the course of studying examples and related theorems. In this example, we are led to believe that there is nothing “structurally” different about P3 and M22. In a certain sense they are the same. Not equal, but the same. One is as good as the other. One is just as interesting as the other.

Here is an extremely basic application of this idea. Suppose we want to compute the following linear combination of polynomials in P3,

5(2 + 3x 4x2 + 5x3) + (3)(3 5x + 3x2 + x3)

Rather than doing it straight-away (which is very easy), we will apply the transformation T to convert into a linear combination of matrices, and then compute in M22 according to the definitions of the vector space operations there (Example VSM),

T 5(2 + 3x 4x2 + 5x3) + (3)(3 5x + 3x2 + x3) = 5T 2 + 3x 4x2 + 5x3 + (3)T 3 5x + 3x2 + x3  Theorem LTLC = 5 510 52 + (3) 23 1 6  Definition of T = 3159 22 8  Operations in M22

Now we will translate our answer back to P3 by applying T1, which we found in Example AIVLT,

T1: M 22P3,T1 ab cd = (acd)+(c+d)x+1 2(abcd)x2+cx3

We compute,

T1 3159 22 8 = 1+30x29x2+22x3

which is, as expected, exactly what we would have computed for the original linear combination had we just used the definitions of the operations in P3 (Example VSP). Notice this is meant only as an illustration and not a suggested route for doing this particular computation.

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 dim U = dim V .

Proof   If U and V are isomorphic, there is an invertible linear transformation T : UV (Definition IVS). T is injective by Theorem ILTIS and so by Theorem ILTD, dim U dim V . Similarly, T is surjective by Theorem ILTIS and so by Theorem SLTD, dim U dim V . The net effect of these two inequalities is that dim U = dim 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 P6 is not isomorphic to M34 since their dimensions (7 and 12, respectively) are not equal. With tools developed in Section VR we will be able to establish that the converse of Theorem IVSED is true. Think about that one for a moment.

Subsection RNLT: 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 T : UV is a linear transformation. Then the rank of T, r T, is the dimension of the range of T,

r T = dim T

(This definition contains Notation ROLT.)

Definition NOLT
Nullity Of a Linear Transformation
Suppose that T : UV is a linear transformation. Then the nullity of T, n T, is the dimension of the kernel of T,

n T = dim KT

(This definition contains Notation NOLT.)

Here are two quick theorems.

Theorem ROSLT
Rank Of a Surjective Linear Transformation
Suppose that T : UV is a linear transformation. Then the rank of T is the dimension of V , r T = dim V , if and only if T is surjective.

Proof   By Theorem RSLT, T is surjective if and only if T = V . Applying Definition ROLT, T = V if and only if r T = dim T = dim V .

Theorem NOILT
Nullity Of an Injective Linear Transformation
Suppose that T : UV is an injective linear transformation. Then the nullity of T is zero, n T = 0, if and only if T is injective.

Proof   By Theorem KILT, T is injective if and only if KT = 0. Applying Definition NOLT, KT = 0 if and only if n T = 0.

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 T : UV is a linear transformation. Then

r T + n T = dim U

Proof   Let r = r T and s = n T. Suppose that R = v1,v2,v3,,vr V is a basis of the range of T, T, and S = u1,u2,u3,,us U is a basis of the kernel of T, KT. Note that R and S are possibly empty, which means that some of the sums in this proof are “empty” and are equal to the zero vector.

Because the elements of R are all in the range of T, each must have a non-empty pre-image by Theorem RPI. Choose vectors wi U, 1 i r such that wi T1 v i. So T wi = vi, 1 i r. Consider the set

B = u1,u2,u3,,us,w1,w2,w3,,wr

We claim that B is a basis for U.

To establish linear independence for B, begin with a relation of linear dependence on B. So suppose there are scalars a1,a2,a3,,as and b1,b2,b3,,br

0 = a1u1 + a2u2 + a3u3 + + asus + b1w1 + b2w2 + b3w3 + + brwr

Then

0 = T 0  Theorem LTTZZ = T a1u1 + a2u2 + a3u3 + + asus+ b1w1 + b2w2 + b3w3 + + brwr  Definition LI = a1T u1 + a2T u2 + a3T u3 + + asT us + b1T w1 + b2T w2 + b3T w3 + + brT wr  Theorem LTLC = a10 + a20 + a30 + + as0+ b1T w1 + b2T w2 + b3T w3 + + brT wr  Definition KLT = 0 + 0 + 0 + + 0+ b1T w1 + b2T w2 + b3T w3 + + brT wr  Theorem ZVSM = b1T w1 + b2T w2 + b3T w3 + + brT wr  Property Z = b1v1 + b2v2 + b3v3 + + brvr  Definition PI

This is a relation of linear dependence on R (Definition RLD), and since R is a linearly independent set (Definition LI), we see that b1 = b2 = b3 = = br = 0. Then the original relation of linear dependence on B becomes

0 = a1u1 + a2u2 + a3u3 + + asus + 0w1 + 0w2 + + 0wr = a1u1 + a2u2 + a3u3 + + asus + 0 + 0 + + 0  Theorem ZSSM = a1u1 + a2u2 + a3u3 + + asus  Property Z

But this is again a relation of linear independence (Definition RLD), now on the set S. Since S is linearly independent (Definition LI), we have a1 = a2 = a3 = = ar = 0. Since we now know that all the scalars in the relation of linear dependence on B must be zero, we have established the linear independence of S through Definition LI.

To now establish that B spans U, choose an arbitrary vector u U. Then T u R(T), so there are scalars c1,c2,c3,,cr such that

T u = c1v1 + c2v2 + c3v3 + + crvr

Use the scalars c1,c2,c3,,cr to define a vector y U,

y = c1w1 + c2w2 + c3w3 + + crwr

Then

T u y = T u T y  Theorem LTLC = T u T c1w1 + c2w2 + c3w3 + + crwr Substitution = T u c1T w1 + c2T w2 + + crT wr  Theorem LTLC = T u c1v1 + c2v2 + c3v3 + + crvr wi T1 v i = T u T u  Substitution = 0  Property AI

So the vector u y is sent to the zero vector by T and hence is an element of the kernel of T. As such it can be written as a linear combination of the basis vectors for KT, the elements of the set S. So there are scalars d1,d2,d3,,ds such that

u y = d1u1 + d2u2 + d3u3 + + dsus

Then

u = u y + y = d1u1 + d2u2 + d3u3 + + dsus + c1w1 + c2w2 + c3w3 + + crwr

This says that for any vector, u, from U, there exist scalars (d1,d2,d3,,ds,c1,c2,c3,,cr) that form u as a linear combination of the vectors in the set B. In other words, B spans U (Definition SS).

So B is a basis (Definition B) of U with s + r vectors, and thus

dim U = s + r = n T + r T

as desired.

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 T : nm defined with the m × n matrix A by T x = Ax. 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 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 T : 66 is a linear transformation and you are able to quickly establish that the kernel is trivial. Then n T = 0. First this means that T is injective by Theorem NOILT. Also, Theorem RPNDD becomes

6 = dim 6 = r T + n T = r T + 0 = r T

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 knowlege 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.

Subsection SLELT: 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. 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.

Archetype D and Archetype E are ideal examples to illustrate connections with linear transformations. Both have the same coefficient matrix,

D = 2 1 7 7 3456 1 1 4 5

To apply the theory of linear transformations to these two archetypes, employ matrix multiplication (Definition MM) and define the linear transformation,

T : 43,T x = Dx = x 1 2 3 1 +x2 1 4 1 +x3 7 5 4 +x4 7 6 5

Theorem MBLT tells us that T is indeed a linear transformation. Archetype D asks for solutions to SD,b, where b = 8 12 4 . In the language of linear transformations this is equivalent to asking for T1 b. In the language of vectors and matrices it asks for a linear combination of the four columns of D that will equal b. One solution listed is w = 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 b,

w + KT = w + zz KT

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 SD,0. Since D has a null space of dimension two, every preimage (and in particular the preimage of 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, d = 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 SD,d is the pre-image of d, T1 d. However, the vector 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, 4, is four, while the codomain, 3, has dimension three. Then

n T = dim 4 r T  Theorem RPNDD = 4 dim T  Definition ROLT 4 3  T subspace of 3 = 1

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 SC,f and the linear transformation S x = Cx. 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.

Subsection READ: Reading Questions

  1. What conditions allow us to easily determine if a linear tranformation is invertible?
  2. What does it mean to say two vector spaces are isomorphic? Both technically, and informally?
  3. How do linear transformations relate to systems of linear equations?

Subsection EXC: Exercises

C10 The archetypes below are linear transformations of the form T : UV that are invertible. For each, the inverse linear transformation is given explicitly as part of the archetype’s description. Verify for each linear transformation that

T1 T = I U T T1 = I V

Archetype R,
Archetype V,
Archetype W  
Contributed by Robert Beezer

C20 Determine if the linear transformation T : P2M22 is (a) injective, (b) surjective, (c) invertible.

T a + bx + cx2 = a + 2b 2c 2a + 2b a + b 4c3a + 2b + 2c

 
Contributed by Robert Beezer Solution [1466]

C21 Determine if the linear transformation S : P3M22 is (a) injective, (b) surjective, (c) invertible.

S a + bx + cx2 + dx3 = a + 4b + c + 2d4a b + 6c d a + 5b 2c + 2d a + 2c + 5d

 
Contributed by Robert Beezer Solution [1467]

C50 Consider the linear transformation S : M12P1 from the set of 1 × 2 matrices to the set of polynomials of degree at most 1, defined by

S ab = (3a+b)+(5a+2b)x

Prove that S is invertible. Then show that the linear transformation

R: P1M12,R r + sx = (2r s)(5r + 3s)

is the inverse of S, that is S1 = R.  
Contributed by Robert Beezer Solution [1470]

M30 The linear transformation S below is invertible. Find a formula for the inverse linear transformation, S1.

S : P1M1,2,S a + bx = 3a + b2a + b

 
Contributed by Robert Beezer Solution [1472]

M31 The linear transformation R: M12M21 is invertible. Determine a formula for the inverse linear transformation R1: M 21M12. (15 points)

R ab = a + 3b 4a + 11b

 
Contributed by Robert Beezer Solution [1474]

T05 Prove that the identity linear transformation (Definition IDLT) is both injective and surjective, and hence invertible.  
Contributed by Robert Beezer

T15 Suppose that T : UV is a surjective linear transformation and dim U = dim V . Prove that T is injective.  
Contributed by Robert Beezer Solution [1476]

T16 Suppose that T : UV is an injective linear transformation and dim U = dim V . Prove that T is surjective.  
Contributed by Robert Beezer

Subsection SOL: Solutions

C20 Contributed by Robert Beezer Statement [1461]
(a) We will compute the kernel of T. Suppose that a + bx + cx2 KT. Then

00 00 = T a + bx + cx2 = a + 2b 2c 2a + 2b a + b 4c3a + 2b + 2c

and matrix equality (Theorem ME) yields the homogeneous system of four equations in three variables,

a + 2b 2c = 0 2a + 2b = 0 a + b 4c = 0 3a + 2b + 2c = 0

The coefficient matrix of this system row-reduces as

1 22 2 2 0 114 3 2 2  RREF 10 2 012 00 0 00 0

From the existence of non-trivial solutions to this system, we can infer non-zero polynomials in KT. By Theorem KILT we then know that T is not injective.

(b) Since 3 = dim P2 < dim M22 = 4, by Theorem SLTD T is not surjective.

(c) Since T is not surjective, it is not invertible by Theorem ILTIS.

C21 Contributed by Robert Beezer Statement [1462]
(a) To check injectivity, we compute the kernel of S. To this end, suppose that a + bx + cx2 + dx3 KS, so

00 00 = S a + bx + cx2 + dx3 = a + 4b + c + 2d4a b + 6c d a + 5b 2c + 2d a + 2c + 5d

this creates the homogeneous system of four equations in four variables,

a + 4b + c + 2d = 0 4a b + 6c d = 0 a + 5b 2c + 2d = 0 a + 2c + 5d = 0

The coefficient matrix of this system row-reduces as,

1 4 1 2 4 1 6 1 1 5 2 2 1 0 2 5  RREF 1000 0100 0010 0001

We recognize the coefficient matrix as being nonsingular, so the only solution to the system is a = b = c = d = 0, and the kernel of S is trivial, KS = 0 + 0x + 0x2 + 0x3. By Theorem KILT, we see that S is injective.

(b) We can establish that S is surjective by considering the rank and nullity of S.

r S = dim P3 n S  Theorem RPNDD = 4 0 = dim M22

So, S is a subspace of M22 (Theorem RLTS) whose dimension equals that of M22. By Theorem EDYES, we gain the set equality S = M22. Theorem RSLT then implies that S is surjective.

(c) Since S is both injective and surjective, Theorem ILTIS says S is invertible.

C50 Contributed by Robert Beezer Statement [1462]
Determine the kernel of S first. The condition that S ab = 0 becomes (3a + b) + (5a + 2b)x = 0 + 0x. Equating coefficients of these polynomials yields the system

3a + b = 0 5a + 2b = 0

This homogeneous system has a nonsingular coefficient matrix, so the only solution is a = 0, b = 0 and thus

KS = 00

By Theorem KILT, we know S is injective. With n S = 0 we employ Theorem RPNDD to find

r S = r S + 0 = r S + n S = dim M12 = 2 = dim P1

Since S P1 and dim S = dim P1, we can apply Theorem EDYES to obtain the set equality S = P1 and therefore S is surjective.

One of the two defining conditions of an invertible linear transformation is (Definition IVLT)

S R a + bx = S R a + bx = S (2a b)(5a + 3b) = 3(2a b) + (5a + 3b) + 5(2a b) + 2(5a + 3b) x = (6a 3b) + (5a + 3b) + (10a 5b) + (10a + 6b) x = a + bx = IP1 a + bx

That R S ab = IM12 ab is similar.

M30 Contributed by Robert Beezer Statement [1463]
Suppose that S1: M 1,2P1 has a form given by

S1 zw = rz + sw+pz + qwx

where r,s,p,q are unknown scalars. Then

a + bx = S1 S a + bx = S1 3a + b2a + b = r(3a + b) + s(2a + b) + p(3a + b) + q(2a + b) x = (3r + 2s)a + (r + s)b + (3p + 2q)a + (p + q)bx

Equating coefficients of these two polynomials, and then equating coefficients on a and b, gives rise to 4 equations in 4 variables,

3r + 2s = 1 r + s = 0 3p + 2q = 0 p + q = 1

This system has a unique solution: r = 1, s = 1, p = 2, q = 3. So the desired inverse linear transformation is

S1 zw = z w+2z + 3wx

Notice that the system of 4 equations in 4 variables could be split into two systems, each with two equations in two variables (and identical coefficient matrices). After making this split, the solution might feel like computing the inverse of a matrix (Theorem CINM). Hmmmm.

M31 Contributed by Robert Beezer Statement [1464]
We are given that R is invertible. The inverse linear transformation can be formulated by considering the pre-image of a generic element of the codomain. With injectivity and surjectivity, we know that the pre-image of any element will be a set of size one — it is this lone element that will be the output of the inverse linear transformation.

Suppose that we set v = x y as a generic element of the codomain, M21. Then if rs = w R1 v,

x y = v = R w = r + 3s 4r + 11s

So we obtain the system of two equations in the two variables r and s,

r + 3s = x 4r + 11s = y

With a nonsingular coefficient matrix, we can solve the system using the inverse of the coefficient matrix,

r = 11x + 3y s = 4x y

So we define,

R1 v = R1 x y = w = rs = 11x + 3y4x y

T15 Contributed by Robert Beezer Statement [1464]
If T is surjective, then Theorem RSLT says T = V , so r T = dim V . In turn, the hypothesis gives r T = dim U. Then, using Theorem RPNDD,

n T = r T + n T r T = dim U dim U = 0

With a null space of zero dimension, KT = 0, and by Theorem KILT we see that T is injective. T is both injective and surjective so by Theorem ILTIS, T is invertible.