Matrix Representation of a linear operator

[*FaerieLight*]

T is a linear operator from the space of 2 by 2 matrices over the complex plane to the complex plane, that is
T: mat(2x2,C)\rightarrowC, given by

T[a b; c d] = a + d

T operates on a 2 by 2 matrix with elements a, b, c, d, in case that isn't entirely clear. So T gives the trace of the matrix, and it can be shown that T is linear.

I'm having trouble finding a matrix representation of T with respect to any basis.
If T can be written as a matrix with respect to a basis, then that matrix applied to the 2 by 2 above would give a 1 by 1 matrix. I don't understand how this can be the case, since you can't multiply a 2 by 2 matrix with any matrix that I know of to get a 1 by 1 matrix as the result.

If I go by the rule that the columns of the matrix are the images of T applied to the basis which you are finding the matrix representation with respect to, then using the basis vectors [1 0; 0 0], [0 1; 0 0], [0 0; 1 0], [0 0; 0 1], which I think are the standard basis vectors for 2 by 2 matrices, I end up with a 1 by 4 matrix of 0s. This doesn't seem right, because I can't multiply a 1 by 4 matrix with a 2 by 2 matrix to get a 1 by 1 matrix.

Can someone please explain to me where I'm going wrong with this?
Thanks

 

[Fredrik]

The matrix will be 1×4, and you won't be multiplying it with any 2×2 matrices, but with a 4×1 matrix.

This should be easy to see if you understand the relationship between linear operators and matrices described in this post.

 

[HallsofIvy]

When you represent a linear operator as a matrix, you don't multiply the matrix by the vector, you multiply the matrix by the representation of the vector in Rⁿ.
That is, you can represent the matrix
\begin{bmatrix}a & b\\ c & d\end{bmatrix}

as the column
\begin{bmatrix}a \\ b \\ c\\ d\end{bmatrix}