6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 2 & - 1 & \alpha
2 & 3 & - 1 & 0
2 & 1 & 2 & - 2
0 & 1 & - 3 & - 2 \end{array} \right)$$ Given that the dimension of the range space of T is 4 , show that \(\alpha \neq 1\). It is now given that \(\alpha = 1\). Show that the vectors $$\left( \begin{array} { l } 1
2
2
0 \end{array} \right) , \quad \left( \begin{array} { l } 2
3
1
1 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } - 1
- 1
2
- 3 \end{array} \right)$$ form a basis for the range space of T . Given also that the vector \(\left( \begin{array} { c } p
1
1
q \end{array} \right)\) is in the range space of T , find a condition satisfied by \(p\) and \(q\).