10 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { H }\), where
$$\mathbf { H } = \left( \begin{array} { r r r r }
1 & 2 & - 3 & - 5
- 1 & 4 & 5 & 1
2 & 3 & 0 & - 3
- 3 & 5 & 7 & 2
\end{array} \right)$$
- Find the dimension of the range space of T .
- Find a basis for the null space of \(T\).
- 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 \(\mathbf { x } = \left( \begin{array} { c } x _ { 1 }
x _ { 2 }
x _ { 3 }
x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)\).]