The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices \(\mathbf { M } _ { 1 }\) and \(\mathbf { M } _ { 2 }\), respectively, where
$$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r }
1 & 1 & 1 & 2
1 & 4 & 7 & 8
1 & 7 & 11 & 13
1 & 2 & 5 & 5
\end{array} \right) , \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r }
2 & 0 & - 1 & - 1
5 & 1 & - 3 & - 3
3 & - 1 & - 1 & - 1
13 & - 1 & - 6 & - 6
\end{array} \right) .$$
- Find a basis for \(R _ { 1 }\), the range space of \(\mathrm { T } _ { 1 }\).
- Find a basis for \(K _ { 2 }\), the null space of \(\mathrm { T } _ { 2 }\), and hence show that \(K _ { 2 }\) is a subspace of \(R _ { 1 }\).
The set of vectors which belong to \(R _ { 1 }\) but do not belong to \(K _ { 2 }\) is denoted by \(W\).
- State whether \(W\) is a vector space, justifying your answer.
The linear transformation \(\mathrm { T } _ { 3 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is the result of applying \(\mathrm { T } _ { 1 }\) and then \(\mathrm { T } _ { 2 }\), in that order.
- Find the dimension of the null space of \(\mathrm { T } _ { 3 }\).