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T: mat(2x2,C)[tex]\rightarrow[/tex]C, 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