Challenging +1.2 This is a straightforward application of the rank-nullity theorem requiring row reduction of a 4×4 matrix to find its rank, then using dimension of null space = 4 - rank. While it involves more computation than typical A-level questions and is from Further Maths, the method is standard and algorithmic with no conceptual subtlety.
1 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix
$$\left( \begin{array} { r r r r }
1 & 5 & 2 & 6 \\
2 & 0 & - 1 & 7 \\
3 & - 1 & - 2 & 10 \\
4 & 10 & 13 & 29
\end{array} \right)$$
Find the dimension of the null space of T .