Let \(V\) be the subspace of \(\mathbb { R } ^ { 4 }\) spanned by
$$\mathbf { v } _ { 1 } = \left( \begin{array} { l }
1 \\
2 \\
0 \\
2
\end{array} \right) , \quad \mathbf { v } _ { 2 } = \left( \begin{array} { r }
- 2 \\
- 5 \\
5 \\
6
\end{array} \right) , \quad \mathbf { v } _ { 3 } = \left( \begin{array} { r }
0 \\
- 3 \\
15 \\
18
\end{array} \right) \quad \text { and } \quad \mathbf { v } _ { 4 } = \left( \begin{array} { r }
0 \\
- 2 \\
10 \\
8
\end{array} \right) .$$
- Show that the dimension of \(V\) is 3 .
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- Express \(\mathbf { v } _ { 4 }\) as a linear combination of \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\) and \(\mathbf { v } _ { 3 }\).
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- Write down a basis for \(V\).
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Let \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 0 & 0 \\ 2 & - 5 & - 3 & - 2 \\ 0 & 5 & 15 & 10 \\ 2 & 6 & 18 & 8 \end{array} \right)\).
- Find the general solution of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\).
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The set of elements of \(\mathbb { R } ^ { 4 }\) which are not solutions of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\) is denoted by \(W\).
- State, with a reason, whether \(W\) is a vector space.
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