The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where
$$\mathbf { M } = \left( \begin{array} { r r r r }
2 & 1 & - 1 & 4
3 & 4 & 6 & 1
- 1 & 2 & 8 & - 7
\end{array} \right)$$
The range space of T is \(R\). In any order,
- show that the dimension of \(R\) is 2 ,
- find a basis for \(R\) and obtain a cartesian equation for \(R\),
- find a basis for the null space of T .
The vector \(\left( \begin{array} { l } 8
7
k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of
$$\mathbf { M } \mathbf { x } = \left( \begin{array} { l }
8
7
k
\end{array} \right) .$$