The archetypes below are all linear transformations whose domains and codomains are vector spaces of column vectors (Definition VSCV). For each one, compute the matrix representation described in the proof of Theorem MLTCV.

Archetype M, Archetype N, Archetype O, Archetype P, Archetype Q, Archetype R

Find the matrix representation of $\ltdefn{T}{\complex{3}}{\complex{4}}$, $\lt{T}{\colvector{x\\y\\z}} = \colvector{3x + 2y + z\\ x + y + z \\ x - 3y \\2x + 3y + z }$.

Let $\vect{w}=\colvector{-3\\1\\4}$. Referring to Example MOLT, compute $\lt{S}{\vect{w}}$ two different ways. First use the definition of $S$, then compute the matrix-vector product $C\vect{w}$ (Definition MVP).

Define the linear transformation \begin{equation*} \ltdefn{T}{\complex{3}}{\complex{2}},\quad \lt{T}{\colvector{x_1\\x_2\\x_3}}=\colvector{2x_1-x_2+5x_3\\-4x_1+2x_2-10x_3} \end{equation*} Verify that $T$ is a linear transformation.

Verify that the function below is a linear transformation. \begin{equation*} \ltdefn{T}{P_2}{\complex{2}},\quad \lt{T}{a+bx+cx^2}=\colvector{2a-b\\b+c} \end{equation*}

Define the linear transformation \begin{equation*} \ltdefn{T}{\complex{3}}{\complex{2}},\quad \lt{T}{\colvector{x_1\\x_2\\x_3}}=\colvector{2x_1-x_2+5x_3\\-4x_1+2x_2-10x_3} \end{equation*} Compute the preimages, $\preimage{T}{\colvector{2\\3}}$ and $\preimage{T}{\colvector{4\\-8}}$.

For the linear transformation $S$ compute the pre-images. \begin{align*} \ltdefn{S}{\complex{3}}{\complex{3}},\quad \lt{S}{\colvector{a\\b\\c}}= \colvector{a-2b-c\\3a-b+2c\\a+b+2c } \end{align*} \begin{align*} \preimage{S}{\colvector{-2\\5\\3}}& & \preimage{S}{\colvector{-5\\5\\7}}& \end{align*}

If $\ltdefn{T}{\complex{2}}{\complex{2}}$ satisfies $\lt{T}{\colvector{2\\1}} = \colvector{3\\4}$ and $\lt{T}{\colvector{1\\1}} = \colvector{-1\\2}$, find $\lt{T}{\colvector{4\\3}}$.

If $\ltdefn{T}{\complex{2}}{\complex{3}}$ satisfies $\lt{T}{\colvector{2\\3}} = \colvector{2\\2\\1}$ and $\lt{T}{\colvector{3\\4}} = \colvector{-1\\0\\2}$, find the matrix representation of $T$.

Define $\ltdefn{T}{M_{22}}{\complex{1}}$ by $\lt{T}{\begin{bmatrix} a & b \\ c & d \end{bmatrix}} = a + b + c - d$. Find the pre-image $\preimage{T}{3}$.

Define $\ltdefn{T}{P_3}{P_2}$ by $\lt{T}{a + bx + cx^2 + dx^3} = b + 2cx + 3dx^2$. Find the pre-image of $\mathbf{0}$. Does this linear transformation seem familiar?

Define two linear transformations, $\ltdefn{T}{\complex{4}}{\complex{3}}$ and $\ltdefn{S}{\complex{3}}{\complex{2}}$ by \begin{align*} \lt{S}{\colvector{x_1\\x_2\\x_3}} &= \colvector{ x_1-2x_2+3x_3\\ 5x_1+4x_2+2x_3 } & \lt{T}{\colvector{x_1\\x_2\\x_3\\x_4}} &= \colvector{ -x_1+3x_2+x_3+9x_4\\ 2x_1+x_3+7x_4\\ 4x_1+2x_2+x_3+2x_4 } \end{align*} Using the proof of Theorem MLTCV compute the matrix representations of the three linear transformations $T$, $S$ and $\compose{S}{T}$. Discover and comment on the relationship between these three matrices.

Suppose $U$ and $V$ are vector spaces and define a function $\ltdefn{Z}{U}{V}$ by $\lt{Z}{\vect{u}}=\zerovector_{V}$ for every $\vect{u}\in U$. Prove that $Z$ is a (stupid) linear transformation. (See Exercise ILT.M60, Exercise SLT.M60, Exercise IVLT.M60.)

Use the conclusion of Theorem LTLC to motivate a new definition of a linear transformation. Then prove that your new definition is equivalent to Definition LT. (Proof Technique D and Proof Technique E might be helpful if you are not sure what you are being asked to prove here.)