The standard matrix [T] of T and if T is invertible

Question

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Answer 1

Hint: The columns of $[T]$ are $T\left(\left[\begin{smallmatrix}1\\0\end{smallmatrix}\right]\right)$ and $T\left(\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\right)$.

Answer 2

The standard matrix of $T$, is $[T]$ such that $$T(v)=w\iff [T]\cdot v=w$$here we have $$T\left(\begin{bmatrix}-3&-2\\2&1\end{bmatrix}\right)=\begin{bmatrix}2&-1\\1&1\end{bmatrix}\iff [T]\begin{bmatrix}-3&-2\\2&1\end{bmatrix}=\begin{bmatrix}2&-1\\1&1\end{bmatrix}$$therefore $$[T]=\begin{bmatrix}2&-1\\1&1\end{bmatrix}\begin{bmatrix}-3&-2\\2&1\end{bmatrix}^{-1}=\begin{bmatrix}2&-1\\1&1\end{bmatrix}\begin{bmatrix}1&2\\-2&-3\end{bmatrix}=\begin{bmatrix}4&7\\-1&-1\end{bmatrix}$$which is invertible and we obtain$$[T^{-1}]=[T]^{-1}={1\over 3}\begin{bmatrix}-1&-7\\1&4\end{bmatrix}$$