Matrix of linear transformation from R^2 to R^{2x2}

[lockedup]

Homework Statement

Find the matrix of the transformation:

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

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

Homework Equations

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

$$\[<br /> \left[ {\begin{array}{cc}<br /> 1 & 0 \\<br /> 0 & 0 \\<br /> 0 & 0 \\<br /> 0 & 1 \\<br /> \end{array} } \right]<br /> \]$$

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?

 

[radou]

If your R^2x2 is the subspace of all 4x4 matrices of this given form, what is its dimension? You have too many basis vectors here.

 

[lockedup]

If your R^2x2 is the subspace of all 4x4 matrices of this given form, what is its dimension? You have too many basis vectors here.

R^2x2 is the space of all 2x2 matrices. It has 4 basis vectors with 1 in the i,j position and zeroes everywhere else.