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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.

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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.

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  • $\begingroup$ So if I understand you correctly, the transformation matrix as given, $\begin{bmatrix}a&b\\0&d\end{bmatrix}$ is in $U^{2 \times 2}$. The challenge then is to find an equivalent $3 \times 3$ transformation matrix in $\mathbb{R}^3$ In other words, contrary to what I was thinking, that M was the 'unknown', an equivalent transformation matrix is the unknown quantity here... if that's the case, I understand. Thank you very much. $\endgroup$ – user10756 Mar 9 '13 at 20:32
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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.

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  • $\begingroup$ Right, thanks much for your input. $\endgroup$ – user10756 Mar 9 '13 at 20:44

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