Let A:
\begin{bmatrix}0&0&0\\0&0&1\\1&2&3\end{bmatrix}
be a matrix associated with the linear mapping T in the basis B={(1,1,1), (0,2,2), (0,0,3)}.
Find the standard matrix for T.
If I'm not misunderstanding:
We want to find the transformation T in the standard basis. I.e. what the transformation would be with the following basis vectors:
\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
Currently the transformation T has done a transformation T on A under the basis B.
I'm not sure how think about it.
Best Answer
Here's how to start thinking about it. I'll use $\mathcal B$ to denote the coordinate representation in the basis $\mathcal B = \left\{(1,1,1),(0,2,2),(0,0,3) \right\}$ and $\mathcal S$ to denote the coordinate representation in the standard basis. The transformation matrix for the basis $\mathcal B$ is given by $$A = \left[ \begin{array}{ccc} 0&0&0 \\ 0&0&1 \\ 1&2&3 \end{array} \right].$$ This tells you that the first basis vector in $\mathcal B$ is mapped to the third basis vector in $\mathcal B$, the second is mapped to two times the third, and so on. That is,
$\left[ \begin{array}{ccc} 0&0&0 \\ 0&0&1 \\ 1&2&3 \end{array} \right]\left[\begin{array}{c}1\\0\\0 \end{array} \right]_{\mathcal B} = \left[\begin{array}{c}0\\0\\1 \end{array} \right]_{\mathcal B} = (1)\left[\begin{array}{c}0\\0\\3 \end{array} \right]_{\mathcal S},$
$\left[ \begin{array}{ccc} 0&0&0 \\ 0&0&1 \\ 1&2&3 \end{array} \right]\left[\begin{array}{c}0\\1\\0 \end{array} \right]_{\mathcal B} = \left[\begin{array}{c}0\\0\\2 \end{array} \right]_{\mathcal B} = (2)\left[\begin{array}{c}0\\0\\3 \end{array} \right]_{\mathcal S}, \text{ and }$
$\left[ \begin{array}{ccc} 0&0&0 \\ 0&0&1 \\ 1&2&3 \end{array} \right]\left[\begin{array}{c}0\\0\\1 \end{array} \right]_{\mathcal B} = \left[\begin{array}{c}0\\1\\3 \end{array} \right]_{\mathcal B} = (1)\left[\begin{array}{c}0\\2\\2 \end{array} \right]_{\mathcal S} + (3)\left[\begin{array}{c}0\\0\\3 \end{array} \right]_{\mathcal S}.$
So you see that in the standard basis \begin{equation} \begin{split} T \text{ maps } & (1,1,1) \text{ to } (0,0,3), \\ T \text{ maps } & (0,2,2) \text{ to } (0,0,6), \\ T \text{ maps } & (0,0,3) \text{ to } (0,2,11). \end{split} \end{equation}
Now see if you can use this information to construct the matrix for $T$ in the standard basis.