The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix
$$\mathbf { M } = \left( \begin{array} { r r r r }
- 1 & 2 & 3 & 4
1 & 0 & 1 & - 1
1 & - 2 & - 3 & a
1 & 2 & 5 & 2
\end{array} \right) .$$
- 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 \(x + 2 y = t\). - For \(a = - 4\), find the general solution of
$$\mathbf { M } \mathbf { x } = \left( \begin{array} { r }
- 1
1
1
1
\end{array} \right)$$
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