The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where
$$\mathbf { M } = \left( \begin{array} { r r r r }
1 & 2 & 3 & 4
1 & - 1 & 2 & 3
1 & - 3 & 3 & 5
1 & 4 & 2 & 2
\end{array} \right)$$
The range space of T is denoted by \(V\).
- Determine the dimension of \(V\).
- Show that the vectors \(\left( \begin{array} { l } 1
1
1
1 \end{array} \right) , \left( \begin{array} { r } 2
- 1
- 3
4 \end{array} \right) , \left( \begin{array} { l } 3
2
3
2 \end{array} \right)\) are a basis of \(V\).
The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\). - State, with a reason, whether \(W\) is a vector space.
- Show that if the vector \(\left( \begin{array} { l } x
y
z
t \end{array} \right)\) belongs to \(W\) then \(x + y \neq z + t\).