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 $V _ { 2 }$.\\
(i) Give a reason why $V _ { 1 } \cup V _ { 2 }$ is not a linear space.\\
(ii) 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.

\hfill \mbox{\textit{CAIE FP1 2007 Q10 [10]}}