10 The vectors \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 } , \mathbf { b } _ { 4 }\) are defined as follows:
$$\mathbf { b } _ { 1 } = \left( \begin{array} { c }
1
0
0
0
\end{array} \right) , \quad \mathbf { b } _ { 2 } = \left( \begin{array} { c }
1
1
0
0
\end{array} \right) , \quad \mathbf { b } _ { 3 } = \left( \begin{array} { c }
1
1
1
0
\end{array} \right) , \quad \mathbf { b } _ { 4 } = \left( \begin{array} { c }
1
1
1
1
\end{array} \right) .$$
The linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 }\) is denoted by \(V _ { 1 }\) and the linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 4 }\) is denoted by \(V _ { 2 }\).
- Give a reason why \(V _ { 1 } \cup V _ { 2 }\) is not a linear space.
- State the dimension of the linear space \(V _ { 1 } \cap V _ { 2 }\) and write down a basis.
Consider now the set \(V _ { 3 }\) of all vectors of the form \(q \mathbf { b } _ { 2 } + r \mathbf { b } _ { 3 } + s \mathbf { b } _ { 4 }\), where \(q , r , s\) are real numbers. Show that \(V _ { 3 }\) is a linear space, and show also that it has dimension 3 .
Determine whether each of the vectors
$$\left( \begin{array} { l }
4
4
2
5
\end{array} \right) \quad \text { and } \quad \left( \begin{array} { l }
5
4
2
5
\end{array} \right)$$
belongs to \(V _ { 3 }\) and justify your conclusions.