Example KVMR concludes with a basis for the kernel of the linear transformation $T$. Compute the value of $T$ for each of these two basis vectors. Did you get what you expected?

Compute the matrix representation of $T$ relative to the bases $B$ and $C$. \begin{align*} \ltdefn{T}{P_3}{\complex{3}},\quad \lt{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}} \end{align*}

Find a matrix representation of the linear transformation $T$ relative to the bases $B$ and $C$. \begin{align*} &\ltdefn{T}{P_2}{\complex{2}},\quad \lt{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}} \end{align*}

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 $\lt{T}{3+5x-2x^2}$. \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} }\\ \lt{T}{a+bx+cx^2}&= \begin{bmatrix} 2a-b+c & a+3b-c \\ a+3b-c & a-c \end{bmatrix} \end{align*}

Use a matrix representation to determine if the linear transformation $\ltdefn{T}{P_3}{M_{22}}$ is surjective. \begin{equation*} \lt{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*}

Find bases for the kernel and range of the linear transformation $S$ below. \begin{equation*} \ltdefn{S}{M_{22}}{P_2},\quad\lt{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*}

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

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

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

Use a matrix representation to find a basis for the range of the linear transformation $L$. \begin{equation*} \ltdefn{L}{M_{22}}{P_2},\quad \lt{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*}

Use a matrix representation to find a basis for the kernel of the linear transformation $L$. \begin{equation*} \ltdefn{L}{M_{22}}{P_2},\quad \lt{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*}

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

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

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

In light of Theorem KNSI and Theorem MRCLT, write a short comparison of Exercise MM.T40 with Exercise ILT.T15.

In light of Theorem RCSI and Theorem MRCLT, write a short comparison of Exercise CRS.T40 with Exercise SLT.T15.

In light of Theorem MRCLT and Theorem IMR, write a short comparison of Theorem SS and Theorem ICLT.

In light of Theorem MRCLT and Theorem IMR, write a short comparison of Theorem NPNT and Exercise IVLT.T40.

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

Theorem VSLT defines the vector space $\vslt{U}{V}$ containing all linear transformations with domain $U$ and codomain $V$. Suppose $\dimension{U}=n$ and $\dimension{V}=m$. Prove that $\vslt{U}{V}$ is isomorphic to $M_{mn}$, 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?)

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

Create an entirely different proof of Theorem IMILT that relies on Definition IVLT to establish the invertibility of $T$, and that relies on Definition MI to establish the invertibility of $A$.

Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are linear transformations, and that $B$, $C$ and $D$ are bases for $U$, $V$, and $W$. Using only Definition MR define matrix representations for $T$ and $S$. 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 $\matrixrep{S}{C}{D}$. Explain how you would use this result to motivate a definition for matrix multiplication that is strikingly similar to Theorem EMP.