The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix
$$\left( \begin{array} { r r r r }
1 & 2 & - 1 & - 1
1 & 3 & - 1 & 0
1 & 0 & 3 & 1
0 & 3 & - 4 & - 1
\end{array} \right) .$$
The range space of T is denoted by \(V\).
- Determine the dimension of \(V\).
- Show that the vectors \(\left( \begin{array} { l } 1
1
1
0 \end{array} \right) , \left( \begin{array} { l } 2
3
0
3 \end{array} \right) , \left( \begin{array} { r } - 1
- 1
3
- 4 \end{array} \right)\) are linearly independent. - Write down 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 \(y - z - t \neq 0\).