Template problems in linear algebra

Let $V$ denote the real vector space of polynomials in $x$ of degree at most 3. Let $\mathcal{B}=\{1, x, x^2, x^3\}$ be a basis for $V$ and $T:V\to V$ be the function defined by $T(f(x))=f(x)+f'(x)$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $T$ is a linear transformation.
	\item Find $[T]_{\mathcal{B}}$, the matrix representation for $T$ in terms of the basis $\mathcal{B}$.
	\item Is $T$ diagonalizable? If yes, find a matrix $A$ so that $A[T]_{\mathcal{B}}A^{-1}$ is diagonal, otherwise explain why $T$ is not diagonalizable.
\end{enumerate}

Let $\operatorname{M}_n({\bf R})$ be the vector space of all $n \times n$ matrices with real entries. We say that $A, B \in \operatorname{M}_n({\bf R})$ commute if $AB = BA$.
\begin{enumerate}[label=\alph*)]
	\item Fix $A \in \operatorname{M}_n({\bf R})$. Prove that the set of all matrices in $\operatorname{M}_n({\bf R})$ that commute with $A$ is a subspace of $\operatorname{M}_n({\bf R})$.
	\item Let $A=\begin{bmatrix} 1 & 1 \\ 1 & 1  \end{bmatrix}\in \operatorname{M}_2({\bf R})$ and let $W\subseteq \operatorname{M}_2({\bf R})$ be the subspace of all matrices of $\operatorname{M}_2({\bf R})$ that commute with $A$. Find a basis of $W$. 
\end{enumerate}

Let $V\subset {\bf R}^5$ be the subspace defined by the equation
\[
	x_1-2x_2+3x_3-4x_4+5x_5=0.
\]
\begin{enumerate}[label=\alph*)]
	\item Find (with justification) a basis for $V$.
	\item Find (with justification) a basis for $V^{\perp}$, the subspace of ${\bf R}^5$ orthogonal to $V$ under the usual dot product.
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation defined by $T\left(\begin{bmatrix} x \\ y \\ z\end{bmatrix}\right) = \begin{bmatrix}  x+y \\ 2z-x \\ y+2z\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Find the matrix that represents $T$ with respect to the standard basis for ${\bf R}^3$.
	\item Find a basis for the kernel of $T$.
	\item Determine the rank of $T$.
\end{enumerate}

Let $A=\begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Determine whether $A$ is diagonalizable, and if so, give its diagonal form along with a diagonalizing matrix.
	\item Compute $A^{42}$. Remember to show all work.
\end{enumerate}

Let $A=\begin{bmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Compute the characteristic polynomial $p_A(x)$ of $A$. It has integer roots.
	\item For each eigenvalue $\lambda$ of $A$, find a basis for the eigenspace $E_{\lambda}$.
	\item Determine if $A$ is diagonalizable. If so, give matrices $P$ and $B$ such that $P^{-1}AP=B$ and $B$ is diagonal. If no, explain carefully why $A$ is not diagonalizable.
\end{enumerate}

Let $A=\begin{bmatrix} 6 & -2 & -1 \\ 10 & -3 & -2 \\ 0 & 0 & 1\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Find bases for the eigenspaces of $A$.
	\item Determine if $A$ is diagonalizable. If so, give an invertible matrix $P$ and diagonal matrix $D$ such that $P^{-1}AP=D$. If not, explain why not.
\end{enumerate}

Let $W\subset {\bf R}^5$ be the subspace spanned by the set of vectors $\{\langle 1,-2,0,2,-1\rangle,\langle -2,4,-1,1,2\rangle,\langle 0,1,2,-2,1\rangle\}$.
\begin{enumerate}[label=(\alph*)]
	\item Compute the dimension of $W$.
	\item Determine the dimension of $W^\perp$, the perpendicular subspace in ${\bf R}^5$.
	\item Find a basis for $W^\perp$.
\end{enumerate}

Let $P_3$ be the real vector space of all real polynomials of degree three or less. Define $L:P_3\to P_3$ by $L(p(x))=p(x)+p(-x)$.
\begin{enumerate}[label=\alph*)]
	\item Prove $L$ is a linear transformation.
	\item Find a basis for the null space of $L$.
	\item Compute the dimension of the image of $L$.
\end{enumerate}

Let $V=\{a_0+a_1\sqrt[3]{2}+a_2\sqrt[3]{4}\mid a_0, a_1, a_2\in {\bf Q}\}\subseteq {\bf R}$. This set is a vector space over ${\bf Q}$.
\begin{enumerate}[label=\alph*)]
	\item Verify $V$ is closed under product (using the usual product operation in ${\bf R}$).
	\item Let $T:V\to V$ be the linear transformation defined by $T(v)=(\sqrt[3]{2}+\sqrt[3]{4}) v$. Find the matrix that represents $T$ with respect to the basis $\{1,\sqrt[3]{2},\sqrt[3]{4}\}$ for $V$.
	\item Determine the characteristic polynomial for $T$.
\end{enumerate}

Suppose $\{{\bf v}_1,{\bf v}_2,{\bf v}_3\}$ is a basis for ${\bf R}^3$ and $T:{\bf R}^3\to {\bf R}^3$ is a linear transformation satisfying the following:
\begin{align*}
	T({\bf v}_1) &= 4{\bf v}_1+2{\bf v}_2\\
	T({\bf v}_2) &= 5{\bf v}_2\\
	T({\bf v}_3) &= -2{\bf v}_1+4{\bf v}_2+5{\bf v}_3.
\end{align*}
Determine the eigenvalues of $T$ and find a basis for each eigenspace.

Let$W\subset {\bf R}^5$ be the space spanned by the vectors
\[
	\left\{\begin{bmatrix} 1 \\ -2 \\ 0 \\ 2 \\ 1\end{bmatrix},\begin{bmatrix} -2 \\ 4 \\ -1 \\ 1 \\ 2\end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 2 \\ -2 \\1\end{bmatrix}\right\}.
\]
\begin{enumerate}[label=\alph*)]
	\item Compute the dimension of $W$.
	\item Let $W^{\perp}=\{{\bf v}\in {\bf R}^5\,\mid\, {\bf v}\cdot {\bf w}=0\text{ for all }w\in W\}$. Determine the dimension of $W^{\perp}$, and explain how this following immediately from (a) using a theorem.
	\item Find a basis for $W^{\perp}$.
\end{enumerate}

Let $L$ be the line in ${\bf R}^2$ defined by $y=-3x$, and let $T:{\bf R}^2\to {\bf R}^2$ be the linear transformation that orthogonally projects onto $L$ and then stretches along $L$ by a factor of two.
\begin{enumerate}[label=\alph*)]
	\item Find the eigenvalues and an eigenbasis $\mathcal{B}$ for $T$.
	\item Determine the matrix for $T$ with respect to the basis $\mathcal{B}$.
	\item Determine the matrix for $T$ with respect to the standard basis.
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the orthogonal projection to a $1$-dimensional linear subspace $L\subset {\bf R}^3$.
\begin{enumerate}[label=\alph*)]
	\item List the eigenvalues of $T$.
	\item Write the characteristic polynomial $p_T(x)$ for $T$.
	\item Is $T$ diagonalizable? Briefly justify your answer.
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the orthogonal projection to a $1$-dimensional linear subspace $L\subset {\bf R}^3$.
\begin{enumerate}[label=\alph*)]
	\item List the eigenvalues of $T$.
	\item Write the characteristic polynomial $p_T(x)$ for $T$.
	\item Is $T$ diagonalizable? Justify your answer.
\end{enumerate}

Let $L$ be the line $L$ parameterized by $L(t)=(2t,-3t,t)$ for $t\in {\bf R}$, and let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that is orthogonal projection onto $L$.
\begin{enumerate}[label=\alph*)]
	\item Describe $\operatorname{ker}(T)$ and $\operatorname{im}(T)$, either implicitly (using equations in $x,y,z$) or parametrically.
	\item List the eigenvalues of $T$ and their geometric multiplicities.
	\item Find a basis for each eigenspace of $T$.
	\item Let $A$ be the matrix for $T$ with respect to the standard basis. Find a diagonal matrix $B$ and an invertible matrix $S$ such that $B=S^{-1}AS$. (You do not have to compute $A$.)
\end{enumerate}

Let $T:{\bf R}^4\to {\bf R}^4$ be orthogonal projection to the $2$-dimensional plane $P$ spanned by the vectors ${\bf v}=(2,0,1,0)$ and ${\bf w}=(-1,0,2,0)$.
\begin{enumerate}[label=\alph*)]
	\item Find (with proof) all eigenvalues and eigenvectors, along with their geometric and algebraic multiplicities.
	\item Find the matrix representing $T$ with respect to the standard basis. Is this matrix diagonalizable? Why or why not?
\end{enumerate}

Let $a, b \in {\bf R}$ and $T: {\bf R}^3 \to {\bf R}^3$ be the linear transformation that is orthogonal projection onto the plane $z=ax+by$ (with respect to the usual Euclidean inner-product on ${\bf R}^3$).
\begin{enumerate}[label=(\alph*)]
	\item Find the eigenvalues of $T$ and bases for the corresponding eigenspaces.
	\item Is $T$ diagonalizable? Justify.
	\item What is the characteristic polynomial of $T$?
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the orthogonal projection onto the plane $z=x+y$, with respect to the standard Euclidean inner product.
	\begin{enumerate}[label=\alph*)]
		\item Write the matrix representation of $T$ with respect to the standard basis.
		
		\item Is $T$ diagonalizable? Justify your answer.
	\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that expands radially by a factor of three around the line parameterized by $L(t)=\begin{bmatrix} 2 \\ 2 \\ -1\end{bmatrix} t$, leaving the line itself fixed (viewed as a subspace).

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Find an eigenbasis for $T$ and provide the matrix representation of $T$ with respect to that basis.
	\item Provide the matrix representation of $T$ with respect to the standard basis.
\end{enumerate}

Let $a,b\in {\bf R}$ and $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation which is reflection across the plane $z=ax+by$.
\begin{enumerate}[label=(\alph*)]
	\item Find the eigenvalues of $T$ and for each find a basis for the corresponding eigenspace.
	\item Is $T$ diagonalizable? Justify.
	\item What is the characteristic polynomial of $T$?
	\item What is the minimal polynomial of $T$?
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that rotates counterclockwise around the $z$-axis by $\frac{2\pi}{3}$.
\begin{enumerate}[label=\alph*)]
	\item Write the matrix for $T$ with respect to the standard basis $\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}\right\}$.
	\item Write the matrix for $T$ with respect to the basis $\left\{\begin{bmatrix} \frac{\sqrt{3}}{2} \\ -\frac{1}{2} \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}\right\}$.
	\item Determine all (complex) eigenvalues of $T$.
	\item Is $T$ diagonalizable over ${\bf C}$? Justify your answer.
\end{enumerate}