$ \begin{bmatrix} \ a & 0 \ \ 0 & b \ \end{bmatrix}$

Show that W is a subspace of space V of all 2X2 matrices.

$ \begin{equation} let, \ P = \begin{bmatrix} p & 0 \ 0 & q \ \end{bmatrix} , R= \begin{bmatrix} r & 0 \ 0 & s \ \end{bmatrix} than \ P+R = \begin{bmatrix} p+r & 0 \ 0 & q+s \ \end{bmatrix} \ \therefore P+Q \ is \ in \ W. \ and, \ kP = k \begin{bmatrix} p & 0 \ 0 & q \ \end{bmatrix} \ kP= \begin{bmatrix} kp & 0 \ 0 & kq \ \end{bmatrix} \ \therefore \text{kP is in W}\ \therefore \text{W is a subspace of V} \end{equation}$