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Define $T: M_{2x2} \rightarrow M_{2x2}$ to be the linear transformation defined by

$$T\left( \begin{bmatrix} a & b\\ c & d\\ \end{bmatrix}\right) = \begin{bmatrix} -2c & 2a+2b-2d\\ -2c & 2c\\ \end{bmatrix}$$

Let $B$ be the standard matrix for $M_{2x2}$ $$B = \left\{\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}\right\}$$

Find the $B$-matrix for $T([T]_B)$


Not sure where to start with this one. Thanks for help in advance.

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  • $\begingroup$ What exactly are you looking for? It looks like you want to apply $T$ to $[T]_B$, but $[T]_B$ is a $4\times 4$ matrix... $\endgroup$ – Ken Duna Apr 27 '17 at 17:34
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I'll go ahead and show you how to find $[T]_B$, but it is unclear what you are looking for here.

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

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

$$T\left( \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \right) = \begin{bmatrix} -2 & 0 \\ -2 & 2 \end{bmatrix}$$

$$T\left( \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right) = \begin{bmatrix} 0 & -2 \\ 0 & 0 \end{bmatrix}_.$$

Therefore the matrix of $T$ with respect to $B$ is:

$$[T]_B = \begin{bmatrix} 0 & 0 & -2 & 0 \\ 2 & 2 & 0 & -2 \\ 0 & 0 & -2 & 0 \\ 0 & 0 & 2 & 0 \end{bmatrix}_.$$

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