10 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { H }$, where

$$\mathbf { H } = \left( \begin{array} { r r r r } 
1 & 2 & - 3 & - 5 \\
- 1 & 4 & 5 & 1 \\
2 & 3 & 0 & - 3 \\
- 3 & 5 & 7 & 2
\end{array} \right)$$

(i) Find the dimension of the range space of T .\\
(ii) Find a basis for the null space of $T$.\\
(iii) It is given that $\mathbf { x }$ satisfies the equation

$$\mathbf { H } \mathbf { x } = \left( \begin{array} { r } 
2 \\
- 10 \\
- 1 \\
- 15
\end{array} \right)$$

Using the fact that

$$\mathbf { H } \left( \begin{array} { r } 
1 \\
- 3 \\
1 \\
- 2
\end{array} \right) = \left( \begin{array} { r } 
2 \\
- 10 \\
- 1 \\
- 15
\end{array} \right) ,$$

find the least possible value of $| \mathbf { x } |$.\\[0pt]
[For the vector $\mathbf { x } = \left( \begin{array} { c } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \\ x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)$.]

\hfill \mbox{\textit{CAIE FP1 2002 Q10 [13]}}