The archetypes below are linear transformations of the form $\ltdefn{T}{U}{V}$ 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 \begin{align*} \compose{\ltinverse{T}}{T}&=I_U & \compose{T}{\ltinverse{T}}&=I_V \end{align*} Archetype R, Archetype V, Archetype W

Determine if the linear transformation $\ltdefn{T}{P_2}{M_{22}}$ is (a) injective, (b) surjective, (c) invertible. \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*}

Determine if the linear transformation $\ltdefn{S}{P_3}{M_{22}}$ is (a) injective, (b) surjective, (c) invertible. \begin{equation*} \lt{S}{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*}

For each linear transformation below: (a) Find the matrix representation of $T$, (b) Calculate $n(T)$, (c) Calculate $r(T)$, (d) Graph the image in either $\real{2}$ or $\real{3}$ as appropriate, (e) How many dimensions are lost?, and (f) How many dimensions are preserved?

1. $\ltdefn{T}{\complex{3}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\x\\x}$
2. $\ltdefn{T}{\complex{3}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\y\\0}$
3. $\ltdefn{T}{\complex{3}}{\complex{2}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\x}$
4. $\ltdefn{T}{\complex{3}}{\complex{2}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\y}$
5. $\ltdefn{T}{\complex{2}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y}} = \colvector{x\\y\\0}$
6. $\ltdefn{T}{\complex{2}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y}} = \colvector{x\\y\\x+y}$

Consider the linear transformation $\ltdefn{S}{M_{12}}{P_1}$ from the set of $1\times 2$ matrices to the set of polynomials of degree at most 1, defined by \begin{equation*} \lt{S}{\begin{bmatrix}a & b\end{bmatrix}}=(3a+b)+(5a+2b)x \end{equation*} Prove that $S$ is invertible. Then show that the linear transformation \begin{equation*} \ltdefn{R}{P_1}{M_{12}},\quad \lt{R}{r+sx}=\begin{bmatrix}(2r-s) & (-5r+3s)\end{bmatrix} \end{equation*} is the inverse of $S$, that is $\ltinverse{S}=R$.

The linear transformation $S$ below is invertible. Find a formula for the inverse linear transformation, $\ltinverse{S}$. \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. 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*}

Rework Example CIVLT, only in place of the basis $B$ for $P_2$, choose instead to use the basis $C=\set{1,\,1+x,\,1+x+x^2}$. This will complicate writing a generic element of the domain of $\ltinverse{T}$ as a linear combination of the basis elements, and the algebra will be a bit messier, but in the end you should obtain the same formula for $\ltinverse{T}$. The inverse linear transformation is what it is, and the choice of a particular basis should not influence the outcome.

Suppose $U$ and $V$ are vector spaces. Define the function $\ltdefn{Z}{U}{V}$ by $\lt{Z}{\vect{u}}=\zerovector_{V}$ for every $\vect{u}\in U$. Then by Exercise LT.M60, $Z$ is a linear transformation. Formulate a condition on $U$ and $V$ that is equivalent to $Z$ being an invertible linear transformation. In other words, fill in the blank to complete the following statement (and then give a proof): $Z$ is invertible if and only if $U$ and $V$ are $\Rule{1in}{1pt}{0pt}$. (See Exercise ILT.M60, Exercise SLT.M60, Exercise MR.M60.)

Prove that the identity linear transformation (Definition IDLT) is both injective and surjective, and hence invertible.

Suppose that $\ltdefn{T}{U}{V}$ is a surjective linear transformation and $\dimension{U}=\dimension{V}$. Prove that $T$ is injective.

Suppose that $\ltdefn{T}{U}{V}$ is an injective linear transformation and $\dimension{U}=\dimension{V}$. Prove that $T$ is surjective.

Suppose that $U$ and $V$ are isomorphic vector spaces, not of dimension zero. Prove that there are infinitely many isomorphisms between $U$ and $V$.

Suppose $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are linear transformations and $\dimension{U}=\dimension{V}=\dimension{W}$. Suppose that $\compose{S}{T}$ is invertible. Prove that $S$ and $T$ are individually invertible (this could be construed as a converse of Theorem CIVLT).