Properties of linear transformations

Find the matrix for the linear transformation that rotates every vector in $\mathbb{R}^{2}$ by an angle of $2\pi/3$ and then reflects about the $x$-axis.

Find the matrix for the linear transformation that rotates every vector in $\mathbb{R}^{2}$ by a counterclockwise angle of $\pi/6$ and then reflects about the $x$-axis followed by a reflection about the $y$-axis.

Find the matrix for the linear transformation that reflects every vector in $\mathbb{R}^{2}$ about the $x$-axis and then rotates by an angle of $\pi/4$.

Find the matrix of the linear transformation that rotates every vector in $\mathbb{R}^{3}$ counterclockwise about the $z$-axis when viewed from the positive $z$-axis by an angle of 30 degrees and then reflects about the $xy$-plane.

Prove the three properties in Proposition 6.27, using only the definition of a linear transformation (i.e., the fact that it preserves addition and scalar multiplication).

Let $T$ be the linear transformation with matrix $A=\left[\begin{array}{rr}3&1\\ -1&2\end{array}\right]$ and $S$ the linear transformation with matrix $B=\left[\begin{array}{rr}0&-2\\ 4&2\end{array}\right]$. Find the matrix of $S\circ T$. Compute $(S\circ T)(\mathbf{v})$ for $\mathbf{v}=\left[\begin{array}{r}2\\ -1\end{array}\right]$.

Let $T$ be a linear transformation and suppose $T\left(\left[\begin{array}{r}1\\ -4\end{array}\right]\right)=\left[\begin{array}{r}2\\ -3\end{array}\right]$. Suppose $S$ is the linear transformation with matrix $B=\left[\begin{array}{rr}1&2\\ -1&3\end{array}\right]$. Find $(S\circ T)(\mathbf{v})$ for $\mathbf{v}=\left[\begin{array}{r}1\\ -4\end{array}\right]$.

What is the inverse of a reflection? Rotation? Shearing? Scaling?

Let $T$ be a linear transformation with matrix $A=\left[\begin{array}{rr}2&1\\ 5&2\end{array}\right]$. Find the matrix of $T^{-1}$.

Let $T$ be the linear transformation given by $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}4x-3y\\ 2x-2y\end{array}\right]$. Find the matrix of $T^{-1}$.