The matrix of a linear transformation

Let $B=\left\{\left[\begin{array}{r}1\\ -1\\ 2\end{array}\right],\left[\begin{array}{r}2\\ 1\\ 2\end{array}\right],\left[\begin{array}{r}-1\\ 0\\ 2\end{array}\right]\right\}$ be a basis of $\mathbb{R}^{3}$ and let $\mathbf{x}=\left[\begin{array}{r}5\\ -1\\ 4\end{array}\right]$ be a vector in $\mathbb{R}^{2}$. Find $\left[\mathbf{x}\right]_{B}$.

Suppose $T:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is a linear transformation such that

Let $E=\left\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\right\}$ be the standard basis of $\mathbb{R}^{3}$, and let

be another basis.

1. Find the matrix of $T$ with respect to $E$, i.e., find $\left[T\right]_{E,E}$.

2. Find $\left[T\right]_{B,B}$.

Let $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a linear transformation defined by

Consider the two bases

and

Find the matrix $M_{B_{2},B_{1}}$ of $T$ with respect to the bases $B_{1}$ and $B_{2}$.

Let $\mathbf{v}=\left[\begin{array}{r}1\\ -2\\ 3\end{array}\right]$ and consider the linear function $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$. Find the matrix of $T$ with respect to the standard basis of $\mathbb{R}^{3}$.

Let $\mathbf{v}=\left[\begin{array}{r}1\\ -2\\ 3\end{array}\right]$ and consider the linear function $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$. Find the matrix of $T$ with respect to the basis

Suppose that $V$ and $W$ are finite-dimensional vector spaces with bases $B$ and $C$, respectively. Let $T:V\to W$ be a linear transformation such that

for $i=1,\ldots,n$. Let $M$ be the matrix whose columns are $\left[\mathbf{v}_{1}\right]_{B},\ldots,\left[\mathbf{v}_{n}\right]_{B}$, and let $N$ be the matrix whose columns are $\left[\mathbf{w}_{1}\right]_{C},\ldots,\left[\mathbf{w}_{n}\right]_{C}$. Suppose that $M$ is invertible. Show that $\left[T\right]_{C,B}=NM^{-1}$.