# Matrix of linear transformation

## Homework Statement

Find the matrix of the transformation:

$$T: R^{2} \rightarrow R^{2x2}$$

$$$T(a,b) = \left[ {\begin{array}{cc} a & 0 \\ 0 & b \\ \end{array} } \right]$$$

## The Attempt at a Solution

I choose the standard bases for $$R^{2}$$ and $$R^{2x2}$$ and call them b and b' respectively.

$$T(1,0) = 1e_{1} + 0e_{2} + 0e_{3} + 0e_{4}$$
$$T(0,1) = 0e_{1} + 0e_{2} + 0e_{3} + 1e_{4}$$

This gives me a matrix of

$$$\left[ {\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \\ \end{array} } \right]$$$

However, this doesn't work when I multiply by the column vector (a,b). I get a column vector of (a, 0, 0, b) instead of a 2x2 matrix. What's going on?