Find matrix for linear transformation L(M) = transpose(M) in the given basis B of 3x3 magic-squares

Question

I already found a basis B for 3x3 magic-squares, but I am unsure of where to start on part b. Finding the transformation matrix for $L(M) = \text{transpose}(M)$ in said basis.

Answer 1

Let's compute the third column of your matrix: let $M_1,M_2,M_3$ denote the matrices of your basis. We find that $$ L(M_3) = \pmatrix{0 & -1 & 1\\1&0&-1\\-1&1&0}. $$ We would like to find (the unique) $x_1,x_2,x_3$ such that $$ L(M_3) = x_1 M_1 + x_2M_2 + x_3 M_3. $$ In this case, we find that $x_1 = 0, x_2 = 0, x_3 = -1$. Thus, the third column of the is $(0,0,-1)$. That is, the matrix of $L$ relative to your basis has the form $$ \pmatrix{?&?&0\\?&?&0\\?&?&-1}. $$