Solution D.C23 Chris Black

The equations specified are equivalent to the system \begin{align*} a + b - c &= 0\\ b + c - d &= 0\\ a - c - d &= 0 \end{align*} The coefficient matrix of this system row-reduces to \begin{align*} \begin{bmatrix} \leading{1} & 0 & 0& -3\\ 0 & \leading{1} & 0 & 1\\ 0 & 0 & \leading{1} & -2 \end{bmatrix} \end{align*} Thus, every solution can be decribed with a suitable choice of $d$, together with $a = 3d$, $b = -d$ and $c = 2d$. Thus the subspace $W$ can be described as \begin{align*} W &= \setparts{\begin{bmatrix} 3d & -d \\ 2d & d\end{bmatrix}}{d\in\complexes} =\spn{\set{\begin{bmatrix} 3 & -1 \\ 2 & 1\end{bmatrix}}} \end{align*} So, $W$ is a subspace of $M_{22}$ with dimension 1.