Exercise 10.4.3 — Matrix Theory and Linear Algebra

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

T\left[\begin{array}{r}1\\ 0\\ 0\end{array}\right]=\left[\begin{array}{r}3\\ 3\\ 3\end{array}\right],\quad T\left[\begin{array}{r}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{r}1\\ 2\\ 3\end{array}\right],\quad T\left[\begin{array}{r}0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{r}1\\ 3\\ -1\end{array}\right].

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

B=\left\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\}=\left\{\left[\begin{array}{r}1\\ 0\\ 0\end{array}\right],~{}\left[\begin{array}{r}1\\ 1\\ 0\end{array}\right],~{}\left[\begin{array}{r}2\\ 1\\ 1\end{array}\right]\right\}

be another basis.

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

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