How to write the matrix operator for finding projection of a matrix along one of the basis matrices?
For example I have a matrix $\mathcal{M}$ which can be written in terms of basis matrices like:
$$a\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} + b\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + c \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $$
How to write the matrix operator which will give me the projection along the direction of (say) $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ ?
P.S: I'm asking this because in quantum computing (for two-qubit systems) the four matrices are written in short-hand as $a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$ or rather $|\psi\rangle_{AB} = a(|0\rangle_A \otimes |0\rangle_B) + b (|0\rangle_A \otimes |1\rangle_B) + c (|1\rangle_A \otimes |0\rangle_B) + d (|1\rangle_A \otimes |1\rangle_B)$? Our professor said that the "measurement operator" can be written as $M_A\otimes I_B$ or $M_B\otimes I_A$ where $M_A$ and $M_B$ are the projection operators for $A$ and $B$ respectively. It is not entirely clear what he meant by projection operator for $A$ or $B$? Projection operator should give the projection along one of the basis states like $|00\rangle, |01\rangle$ etc. no? So, what does projection operator for $A$ or projection operator for $B$ even mean in this case? $I_A$ and $I_B$ as far as I understand are the identity operators which when act on any state of qubit $A$ or qubit $B$, returns the same state.
Note: I have represented $|0\rangle$ by the matrix \begin{bmatrix} 1 \\ 0 \end{bmatrix} and $|1\rangle$ by the matrix \begin{bmatrix} 0 \\ 1 \end{bmatrix}