# 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

## 1 Answer

It seems that you have found a right transformation (didn't compute it myself but if you did the same for the rest of the vectors you should be fine) note that your transformation matrix uses coordinate vectors, what is the question then?

• Sorry should've clarified; I am looking for the transformation matrix of T using the basis E and the linear transformation listed above. – user300011 Feb 14 '16 at 23:48
• 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
• You got the Ax=b form of a linear transformation. That should be enough. :) – Pavel Penshin Feb 14 '16 at 23:58