Find the matrix of the given linear transformation $T$. Here's the specific scenario:
$T(M) = \begin{bmatrix}1&2\\0&3\end{bmatrix}M$ from  $U^{2 \times 2}$ to $U^{2 \times 2}$ with respect to $\mathfrak{B} =\left(\begin{bmatrix}1&0\\0&0\end{bmatrix},\begin{bmatrix}0&1\\0&0\end{bmatrix},\begin{bmatrix}0&1\\0&1\end{bmatrix}\right)$
Now I probably can figure out how to fudge a correct answer to this, so that's not the problem.  The problem is I don't understand what the question is asking... Using a rough analog w/o matrices, I read this question as follows:
$f(x) = 5x$; $\forall x,f(x) \in \mathbb{R}$. Coordinate system: polar.  Find $x$.
Find $x$ -- or in the given case, $M$?  I'm used to specifying an independent variable in order to find a dependent one; so, needless to say, this scenario is completely upside down given my normal frame of reference.  Clearly, I'm not thinking about this in the right way -- would appreciate any insight readers of this query can provide.  Thanks much in advance.         
 A: Given an upper triangular matrix, for example
$$\begin{bmatrix} 4 & 2 \\ 0 & 3\end{bmatrix}$$
we can write this as a linear combination of the matrices is $\mathfrak B$:
$$4\begin{bmatrix} 1 & 0 \\ 0 & 0\end{bmatrix} - \begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix} + 3\begin{bmatrix} 0 & 1 \\ 0 & 1\end{bmatrix}$$
so we can think of the matrix as being represented by the column vector
$$\begin{bmatrix} 4 \\ -1 \\ 3\end{bmatrix}$$
which contains the coefficients from that linear combination.
This means we are using the basis to identify $U^{2 \times 2}$ with $\mathbb R^3$.  The map $T\colon U^{2 \times 2} \to U^{2 \times 2}$ can then be thought of as a map $T\colon\mathbb R^3 \to \mathbb R^3$.  We know such maps can be given as multiplication by a $3 \times 3$ matrix.  The question asks you to find that matrix.
A: You have a basis for $U^{2 \times 2}$: 
$$e_1 = \left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right]$$
$$e_2 = \left[\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right]$$
$$e_3 = \left[\begin{array}{cc} 0 & 1 \\ 0 & 1 \end{array}\right]$$
Now you want to figure out what $T$ does to each of the basis vectors. For example:
$$T(e_1) = \left[\begin{array}{cc} 1 & 2 \\ 0 & 3 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right] = \left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right] = e_1$$
You can repeat this for $e_2$ and $e_3$, and this allows you to write $T$ as a $3 \times 3$ matrix.
