Finding matrix from linear transformation

Question

Let $T : R^2 → R^2$ be the linear transformation defined by

$T(\begin{bmatrix}1\\2\end{bmatrix})=\begin{bmatrix}2\\1\end{bmatrix}$

$T(\begin{bmatrix}1\\1\end{bmatrix})=\begin{bmatrix}0\\1\end{bmatrix}$

Find the matrix of T with respect to the standard basis $E_2$ = {$\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}0\\1\end{bmatrix}$}

How do I begin to solve for the matrix T? Am I supposed to use the basis $E_2$ to get from $T(\begin{bmatrix}1\\2\end{bmatrix}) to \begin{bmatrix}2\\1\end{bmatrix}$ and so on?

Answer 1

Hint:

$$T\left(\pmatrix{0 \\1}\right)=T\left(\pmatrix{1 \\2}\right)-T\left(\pmatrix{1 \\1}\right)$$

$$T\left(\pmatrix{1 \\0}\right)=T\left(\pmatrix{1 \\1}\right)-T\left(\pmatrix{0 \\1}\right)$$

Answer 2

Guide:$$T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) + 2 T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix}\right) = \begin{bmatrix}2 \\ 1 \end{bmatrix}$$

$$T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) + T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix}\right) = \begin{bmatrix}0 \\ 1 \end{bmatrix}$$

We should be able to solve for $T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix}\right)$ and $T \left( \begin{bmatrix} 1\\ 0 \end{bmatrix}\right)$