The matrix \(\mathbf { A }\), given by
$$\mathbf { A } = \left( \begin{array} { r r r r }
1 & - 1 & 0 & 2
3 & - 1 & 4 & 0
5 & - 8 & - 6 & 19
- 2 & 3 & 2 & - 7
\end{array} \right) ,$$
represents a transformation from \(\mathbb { R } ^ { 4 }\) to \(\mathbb { R } ^ { 4 }\).
- Find the rank of \(\mathbf { A }\) and show that \(\left\{ \left( \begin{array} { r } 2
2
- 1
0 \end{array} \right) , \left( \begin{array} { l } 1
3
0
1 \end{array} \right) \right\}\) is a basis for the null space of the transformation. - Show that if
$$\mathbf { A x } = p \left( \begin{array} { r }
1
3
5
- 2
\end{array} \right) + q \left( \begin{array} { r }
- 1
- 1
- 8
3
\end{array} \right) ,$$
where \(p\) and \(q\) are given real numbers, then
$$\mathbf { x } = \left( \begin{array} { c }
p + 2 \lambda + \mu
q + 2 \lambda + 3 \mu
- \lambda
\mu
\end{array} \right) ,$$
where \(\lambda\) and \(\mu\) are real numbers. - Find the values of \(p\) and \(q\) such that
$$p \left( \begin{array} { r }
1
3
5
- 2
\end{array} \right) + q \left( \begin{array} { r }
- 1
- 1
- 8
3
\end{array} \right) = \left( \begin{array} { r }
3
7