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SectionMRMatrix Representations

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (Theorem MBLT, Theorem MLTCV). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

SubsectionMRMatrix Representations

This is a fundamental definition.

DefinitionMRMatrix Representation

Suppose that \(\ltdefn{T}{U}{V}\) is a linear transformation, \(B=\set{\vectorlist{u}{n}}\) is a basis for \(U\) of size \(n\text{,}\) and \(C\) is a basis for \(V\) of size \(m\text{.}\) Then the matrix representation of \(T\) relative to \(B\) and \(C\) is the \(m\times n\) matrix, \begin{equation*} \matrixrep{T}{B}{C}=\left[ \left.\vectrep{C}{\lteval{T}{\vect{u}_1}}\right| \left.\vectrep{C}{\lteval{T}{\vect{u}_2}}\right| \left.\vectrep{C}{\lteval{T}{\vect{u}_3}}\right| \ldots \left|\vectrep{C}{\lteval{T}{\vect{u}_n}}\right. \right]\text{.} \end{equation*}

We may choose to use whatever terms we want when we make a definition. Some are arbitrary, while others make sense, but only in light of subsequent theorems. Matrix representation is in the latter category. We begin with a linear transformation and produce a matrix. So what? Here is the theorem that justifies the term matrix representation.

Proof

This theorem says that we can apply \(T\) to \(\vect{u}\) and coordinatize the result relative to \(C\) in \(V\text{,}\) or we can first coordinatize \(\vect{u}\) relative to \(B\) in \(U\text{,}\) then multiply by the matrix representation. Either way, the result is the same. So the effect of a linear transformation can always be accomplished by a matrix-vector product (Definition MVP). That is important enough to say again. The effect of a linear transformation is a matrix-vector product.

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Figure8.24Fundamental Theorem of Matrix Representations

The alternative conclusion of this result might be even more striking. It says that to effect a linear transformation (\(T\)) of a vector (\(\vect{u}\)), coordinatize the input (with \(\vectrepname{B}\)), do a matrix-vector product (with \(\matrixrep{T}{B}{C}\)), and un-coordinatize the result (with \(\vectrepinvname{C}\)). So, absent some bookkeeping about vector representations, a linear transformation is a matrix. To adjust the diagram, we “reverse” the arrow on the right, which means inverting the vector representation \(\vectrepname{C}\) on \(V\text{.}\) Now we can go directly across the top of the diagram, computing the linear transformation between the abstract vector spaces. Or, we can around the other three sides, using vector representation, a matrix-vector product, followed by un-coordinatization.

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Figure8.25Fundamental Theorem of Matrix Representations (Alternate)

Here is an example to illustrate how the “action” of a linear transformation can be effected by matrix multiplication.

We will use Theorem FTMR frequently in the next few sections. A typical application will feel like the linear transformation \(T\) “commutes” with a vector representation, \(\vectrepname{C}\text{,}\) and as it does the transformation morphs into a matrix, \(\matrixrep{T}{B}{C}\text{,}\) while the vector representation changes to a new basis, \(\vectrepname{B}\text{.}\) Or vice-versa.

SubsectionNRFONew Representations from Old

In Subsection LT.NLTFO we built new linear transformations from other linear transformations. Sums, scalar multiples and compositions. These new linear transformations will have matrix representations as well. How do the new matrix representations relate to the old matrix representations? Here are the three theorems.

Proof
Proof

The vector space of all linear transformations from \(U\) to \(V\) is now isomorphic to the vector space of all \(m\times n\) matrices.

Proof

This is the second great surprise of introductory linear algebra. Matrices are linear transformations (functions, really), and matrix multiplication is function composition! We can form the composition of two linear transformations, then form the matrix representation of the result. Or we can form the matrix representation of each linear transformation separately, then multiply the two representations together via Definition MM. In either case, we arrive at the same result.

A diagram, similar to ones we have seen earlier, might make the importance of this theorem clearer.

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Figure8.32Matrix Representation and Composition of Linear Transformations

One of our goals in the first part of this book is to make the definition of matrix multiplication (Definition MVP, Definition MM) seem as natural as possible. However, many of us are brought up with an entry-by-entry description of matrix multiplication (Theorem EMP) as the definition of matrix multiplication, and then theorems about columns of matrices and linear combinations follow from that definition. With this unmotivated definition, the realization that matrix multiplication is function composition is quite remarkable. It is an interesting exercise to begin with the question, “What is the matrix representation of the composition of two linear transformations?” and then, without using any theorems about matrix multiplication, finally arrive at the entry-by-entry description of matrix multiplication. Try it yourself (Exercise MR.T80).

SubsectionPMRProperties of Matrix Representations

It will not be a surprise to discover that the kernel and range of a linear transformation are closely related to the null space and column space of the transformation's matrix representation. Perhaps this idea has been bouncing around in your head already, even before seeing the definition of a matrix representation. However, with a formal definition of a matrix representation (Definition MR), and a fundamental theorem to go with it (Theorem FTMR) we can be formal about the relationship, using the idea of isomorphic vector spaces (Definition IVS). Here are the twin theorems.

Proof

An entirely similar result applies to the range of a linear transformation and the column space of a matrix representation of the linear transformation.

Proof

Theorem KNSI and Theorem RCSI can be viewed as further formal evidence for the The Coordinatization Principle, though they are not direct consequences.

Figure 8.38 is meant to suggest Theorem KNSI and Theorem RCSI, in addition to their proofs (and so carry the same notation as the statements of these two theorems). The dashed lines indicate a subspace relationship, with the smaller vector space lower down in the diagram. The central square is highly reminiscent of Figure 8.24. Each of the four vector representations is an isomorphism, so the inverse linear transformation could be depicted with an arrow pointing in the other direction. The four vector spaces across the bottom are familiar from the earliest days of the course, while the four vector spaces across the top are completely abstract. The vector representations that are restrictions (far left and far right) are the functions shown to be invertible representations as the key technique in the proofs of Theorem KNSI and Theorem RCSI. So this diagram could be helpful as you study those two proofs.

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Figure8.38Kernel and Range Isomorphisms

SubsectionIVLTInvertible Linear Transformations

We have seen, both in theorems and in examples, that questions about linear transformations are often equivalent to questions about matrices. It is the matrix representation of a linear transformation that makes this idea precise. Here is our final theorem that solidifies this connection.

Proof

By now, the connections between matrices and linear transformations should be starting to become more transparent, and you may have already recognized the invertibility of a matrix as being tantamount to the invertibility of the associated matrix representation. The next example shows how to apply this theorem to the problem of actually building a formula for the inverse of an invertible linear transformation.

You might look back at Example AIVLT, where we first witnessed the inverse of a linear transformation and recognize that the inverse (\(S\)) was built from using the method of Example ILTVR with a matrix representation of \(T\text{.}\)

Proof

This theorem may seem gratuitous. Why state such a special case of Theorem IMR? Because it adds another condition to our NMEx series of theorems, and in some ways it is the most fundamental expression of what it means for a matrix to be nonsingular — the associated linear transformation is invertible. This is our final update.

Proof

SubsectionReading Questions

2

Find the matrix representation, \(\matrixrep{T}{B}{C}\) of the linear transformation \begin{equation*} \ltdefn{T}{\complex{2}}{\complex{2}},\quad\lteval{T}{\colvector{x_1\\x_2}}=\colvector{2x_1-x_2\\3x_1+2x_2} \end{equation*} relative to the bases \begin{align*} B&=\set{\colvector{2\\3},\,\colvector{-1\\2}}& C&=\set{\colvector{1\\0},\,\colvector{1\\1}}\text{.} \end{align*}

3

What is the second “surprise,” and why is it surprising?

SubsectionExercises

C10

Example KVMR concludes with a basis for the kernel of the linear transformation \(T\text{.}\) Compute the value of \(T\) for each of these two basis vectors. Did you get what you expected?

C20

Compute the matrix representation of \(T\) relative to the bases \(B\) and \(C\text{,}\) \begin{gather*} \ltdefn{T}{P_3}{\complex{3}},\quad \lteval{T}{a+bx+cx^2+dx^3}= \colvector{2a-3b+4c-2d\\a+b-c+d\\3a+2c-3d}\\ B=\set{1,\,x,\,x^2,\,x^3}\quad\quad C=\set{\colvector{1\\0\\0},\,\colvector{1\\1\\0},\,\colvector{1\\1\\1}}\text{.} \end{gather*}

Solution
C21

Find a matrix representation of the linear transformation \(T\) relative to the bases \(B\) and \(C\text{,}\) \begin{align*} &\ltdefn{T}{P_2}{\complex{2}},\quad \lteval{T}{p(x)}=\colvector{p(1)\\p(3)}\\ &B=\set{2-5x+x^2,\,1+x-x^2,\,x^2}\\ &C=\set{\colvector{3\\4},\,\colvector{2\\3}}\text{.} \end{align*}

Solution
C22

Let \(S_{22}\) be the vector space of \(2\times 2\) symmetric matrices. Build the matrix representation of the linear transformation \(\ltdefn{T}{P_2}{S_{22}}\) relative to the bases \(B\) and \(C\) and then use this matrix representation to compute \(\lteval{T}{3+5x-2x^2}\text{,}\) \begin{align*} B&=\set{1,\,1+x,\,1+x+x^2} & C&=\set{ \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix},\, \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix},\, \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} }\\ \lteval{T}{a+bx+cx^2}&= \begin{bmatrix} 2a-b+c & a+3b-c \\ a+3b-c & a-c \end{bmatrix}\text{.} \end{align*}

Solution
C25

Use a matrix representation to determine if the linear transformation \(\ltdefn{T}{P_3}{M_{22}}\) is surjective. \begin{equation*} \lteval{T}{a+bx+cx^2+dx^3}= \begin{bmatrix} -a+4b+c+2d & 4a-b+6c-d\\ a+5b-2c+2d & a+2c+5d \end{bmatrix} \end{equation*}

Solution
C30

Find bases for the kernel and range of the linear transformation \(S\) below. \begin{equation*} \ltdefn{S}{M_{22}}{P_2},\quad\lteval{S}{\begin{bmatrix}a&b\\c&d\end{bmatrix}}= (a+2b+5c-4d)+(3a-b+8c+2d)x+(a+b+4c-2d)x^2 \end{equation*}

Solution
C40

Let \(S_{22}\) be the set of \(2\times 2\) symmetric matrices. Verify that the linear transformation \(R\) is invertible and find \(\ltinverse{R}\text{.}\) \begin{equation*} \ltdefn{R}{S_{22}}{P_2},\quad\lteval{R}{\begin{bmatrix}a&b\\b&c\end{bmatrix}}= (a-b)+(2a-3b-2c)x+(a-b+c)x^2 \end{equation*}

Solution
C41

Prove that the linear transformation \(S\) is invertible. Then find a formula for the inverse linear transformation, \(\ltinverse{S}\text{,}\) by employing a matrix inverse. \begin{equation*} \ltdefn{S}{P_1}{M_{12}},\quad \lteval{S}{a+bx}= \begin{bmatrix} 3a+b & 2a+b \end{bmatrix} \end{equation*}

Solution
C42

The linear transformation \(\ltdefn{R}{M_{12}}{M_{21}}\) is invertible. Use a matrix representation to determine a formula for the inverse linear transformation \(\ltdefn{\ltinverse{R}}{M_{21}}{M_{12}}\text{.}\) \begin{equation*} \lteval{R}{\begin{bmatrix}a & b\end{bmatrix}} = \begin{bmatrix} a+3b\\ 4a+11b \end{bmatrix} \end{equation*}

Solution
C50

Use a matrix representation to find a basis for the range of the linear transformation \(L\text{.}\) \begin{equation*} \ltdefn{L}{M_{22}}{P_2},\quad \lteval{T}{\begin{bmatrix}a&b\\c&d\end{bmatrix}}= (a+2b+4c+d)+(3a+c-2d)x+(-a+b+3c+3d)x^2 \end{equation*}

Solution
C51

Use a matrix representation to find a basis for the kernel of the linear transformation \(L\text{.}\) \begin{equation*} \ltdefn{L}{M_{22}}{P_2},\quad \lteval{T}{\begin{bmatrix}a&b\\c&d\end{bmatrix}}= (a+2b+4c+d)+(3a+c-2d)x+(-a+b+3c+3d)x^2 \end{equation*}

C52

Find a basis for the kernel of the linear transformation \(\ltdefn{T}{P_2}{M_{22}}\text{.}\) \begin{equation*} \lteval{T}{a+bx+cx^2}= \begin{bmatrix} a+2b-2c & 2a+2b \\ -a+b-4c & 3a+2b+2c \end{bmatrix} \end{equation*}

Solution
M20

The linear transformation \(D\) performs differentiation on polynomials. Use a matrix representation of \(D\) to find the rank and nullity of \(D\text{.}\) \begin{equation*} \ltdefn{D}{P_n}{P_n},\quad \lteval{D}{p(x)}=p^\prime(x) \end{equation*}

Solution
M60

Suppose \(U\) and \(V\) are vector spaces and define a function \(\ltdefn{Z}{U}{V}\) by \(\lteval{Z}{\vect{u}}=\zerovector_{V}\) for every \(\vect{u}\in U\text{.}\) Then Exercise IVLT.M60 asks you to formulate the theorem: \(Z\) is invertible if and only if \(U=\set{\zerovector_U}\) and \(V=\set{\zerovector_V}\text{.}\) What would a matrix representation of \(Z\) look like in this case? How does Theorem IMR read in this case?

T20

Construct a new solution to Exercise B.T50 along the following outline. From the \(n\times n\) matrix \(A\text{,}\) construct the linear transformation \(\ltdefn{T}{\complex{n}}{\complex{n}}\text{,}\) \(\lteval{T}{\vect{x}}=A\vect{x}\text{.}\) Use Theorem NI, Theorem IMILT and Theorem ILTIS to translate between the nonsingularity of \(A\) and the surjectivity/injectivity of \(T\text{.}\) Then apply Theorem ILTB and Theorem SLTB to connect these properties with bases.

Solution
T40

Theorem VSLT defines the vector space \(\vslt{U}{V}\) containing all linear transformations with domain \(U\) and codomain \(V\text{.}\) Suppose \(\dimension{U}=n\) and \(\dimension{V}=m\text{.}\) Prove that \(\vslt{U}{V}\) is isomorphic to \(M_{mn}\text{,}\) the vector space of all \(m\times n\) matrices (Example VSM). (Hint: we could have suggested this exercise in Chapter LT, but have postponed it to this section. Why?)

T41

Theorem VSLT defines the vector space \(\vslt{U}{V}\) containing all linear transformations with domain \(U\) and codomain \(V\text{.}\) Determine a basis for \(\vslt{U}{V}\text{.}\) (Hint: study Exercise MR.T40 first.)

T80

Suppose that \(\ltdefn{T}{U}{V}\) and \(\ltdefn{S}{V}{W}\) are linear transformations, and that \(B\text{,}\) \(C\) and \(D\) are bases for \(U\text{,}\) \(V\text{,}\) and \(W\text{.}\) Using only Definition MR define matrix representations for \(T\) and \(S\text{.}\) Using these two definitions, and Definition MR, derive a matrix representation for the composition \(\compose{S}{T}\) in terms of the entries of the matrices \(\matrixrep{T}{B}{C}\) and \(\matrixrep{S}{C}{D}\text{.}\) Explain how you would use this result to motivate a definition for matrix multiplication that is strikingly similar to Theorem EMP.

Solution