The linear transformations $\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ and $\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ are represented by the matrices $\mathbf { M } _ { 1 }$ and $\mathbf { M } _ { 2 }$, respectively, where

$$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 
1 & 1 & 1 & 2 \\
1 & 4 & 7 & 8 \\
1 & 7 & 11 & 13 \\
1 & 2 & 5 & 5
\end{array} \right) , \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 
2 & 0 & - 1 & - 1 \\
5 & 1 & - 3 & - 3 \\
3 & - 1 & - 1 & - 1 \\
13 & - 1 & - 6 & - 6
\end{array} \right) .$$

(i) Find a basis for $R _ { 1 }$, the range space of $\mathrm { T } _ { 1 }$.\\
(ii) Find a basis for $K _ { 2 }$, the null space of $\mathrm { T } _ { 2 }$, and hence show that $K _ { 2 }$ is a subspace of $R _ { 1 }$.

The set of vectors which belong to $R _ { 1 }$ but do not belong to $K _ { 2 }$ is denoted by $W$.\\
(iii) State whether $W$ is a vector space, justifying your answer.

The linear transformation $\mathrm { T } _ { 3 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is the result of applying $\mathrm { T } _ { 1 }$ and then $\mathrm { T } _ { 2 }$, in that order.\\
(iv) Find the dimension of the null space of $\mathrm { T } _ { 3 }$.

\hfill \mbox{\textit{CAIE FP1 2009 Q12 OR}}