Finding a transformation matrix given a basis of matrices

I am looking for the transformation matrix of T using the basis E and the linear transformation listed below. I'm not confident that the way I am solving the problem is the correct way and if the final answer is correct.

I am given the following linear transformation: $$T\left( \begin{bmatrix} a & b \\ 0 & d \\ \end{bmatrix}\right) = \begin{bmatrix} 1 & 2 \\ 0 & 3 \\ \end{bmatrix}\begin{bmatrix} a & b \\ 0 & d \\ \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 0 & 3 \\ \end{bmatrix}\$$

And I have the following basis E:

$$E=\left(e_1=\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}, e_2=\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix},e_3=\begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix}\right)$$

Now to find the transformation matrix $[T]_E^E$ I have to do the following: $$T\left( \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}\right)=\begin{bmatrix} 1 & 2 \\ 0 & 3 \\ \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 0 & 3 \\ \end{bmatrix}$$

and I get $$\begin{bmatrix} 1 & 2 \\ 0 & 0 \\ \end{bmatrix}=1e_1 + 2e_2 + 0e_3$$

Now does this mean I get the vector

$$\begin{bmatrix} 1 \\ 2 \\ 0 \\ \end{bmatrix}$$

and the final result is the following 3x3 matrix:

\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 6\\ 0 & 0 & 9 \\ \end{bmatrix}

Or am I just confused and nowhere near the correct method?

• Are $e_i$'s vectors or matrices? – Pavel Feb 14 '16 at 23:23
• They're matrices guess I got too trigger happy with jax – user300011 Feb 14 '16 at 23:24
• @Pavel should be fixed now – user300011 Feb 14 '16 at 23:28
• And you write matrix as a vector? I am a bit confused by the formulation of the question... – Pavel Feb 14 '16 at 23:33
• I'm not confident as to what I'm doing. I'm looking for a transformation matrix $[T]_E^E$ – user300011 Feb 14 '16 at 23:35

• Say you have an arbitrary linear transformation which is originally represented by some equation in your vector space subspace (in your case $2x2$ matrices which have 3 components, a 3 dimensional subspace.) You have performed the linear transformation on each basis vector in your subspace...(matrix in our case). And then transformed your solutions into a corresponding coordinate vectors in R³. You got a matrix, lets name it A. This matrix is a" representation of your transformation this representation holds true in the coordinate apace – Pavel Penshin Feb 14 '16 at 23:54