The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix

$$\mathbf { M } = \left( \begin{array} { r r r r } 
- 1 & 2 & 3 & 4 \\
1 & 0 & 1 & - 1 \\
1 & - 2 & - 3 & a \\
1 & 2 & 5 & 2
\end{array} \right) .$$

(i) For $a \neq - 4$, the range space of T is denoted by $V$.\\
(a) Find the dimension of $V$ and show that

$$\left( \begin{array} { r } 
- 1 \\
1 \\
1 \\
1
\end{array} \right) , \quad \left( \begin{array} { r } 
2 \\
0 \\
- 2 \\
2
\end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 
4 \\
- 1 \\
a \\
2
\end{array} \right)$$

form a basis for $V$.\\

(b) Show that if $\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)$ belongs to $V$ then $x + 2 y = t$.\\

(ii) For $a = - 4$, find the general solution of

$$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } 
- 1 \\
1 \\
1 \\
1
\end{array} \right)$$

If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q11 OR}}