| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Range space basis and dimension |
| Difficulty | Challenging +1.8 This question requires understanding of range space, rank-nullity theorem, linear independence verification, and vector space axioms. While the techniques are standard Further Maths content (row reduction, linear independence testing), it demands multiple conceptual steps including recognizing column vectors form the range space, understanding that W (complement of a subspace) isn't a vector space, and deriving a constraint equation. The multi-part structure and conceptual depth place it above average difficulty for Further Maths. |
| Spec | 4.03a Matrix language: terminology and notation4.03s Consistent/inconsistent: systems of equations |
The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix
$$\left( \begin{array} { r r r r }
1 & 2 & - 1 & - 1 \\
1 & 3 & - 1 & 0 \\
1 & 0 & 3 & 1 \\
0 & 3 & - 4 & - 1
\end{array} \right) .$$
The range space of T is denoted by $V$.\\
(i) Determine the dimension of $V$.\\
(ii) Show that the vectors $\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } 2 \\ 3 \\ 0 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \\ - 4 \end{array} \right)$ are linearly independent.\\
(iii) Write down a basis of $V$.
The set of elements of $\mathbb { R } ^ { 4 }$ which do not belong to $V$ is denoted by $W$.\\
(iv) State, with a reason, whether $W$ is a vector space.\\
(v) Show that if the vector $\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)$ belongs to $W$ then $y - z - t \neq 0$.
\hfill \mbox{\textit{CAIE FP1 2008 Q12 OR}}