In the inner product space $M_{2\times2}$ with $A\cdot B = \operatorname{trace}(A^\top B)$, find the projection from $A$ to $B$: $$ A=\pmatrix{1&2\\ 3&4},\ B=\pmatrix{1&0\\ 0&0}. $$
Do you just solve it as a regular projection? $$ \operatorname{proj}_B(A) = \frac{A\cdot B}{\|B\|^2} B. $$ Would you just get the matrix $\pmatrix{1&0\\ 0&0}$?