| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| 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, basis, dimension, and the rank-nullity theorem. Part (i) needs row reduction to find rank. Part (ii) requires showing linear independence and span. Parts (iii-iv) involve deeper conceptual understanding of vector spaces and complements. While systematic, it demands multiple advanced techniques and conceptual maturity beyond standard A-level, typical of Further Maths linear algebra. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03l Singular/non-singular matrices4.03r Solve simultaneous equations: using inverse matrix |
The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { M }$, where
$$\mathbf { M } = \left( \begin{array} { r r r r }
1 & 2 & 3 & 4 \\
1 & - 1 & 2 & 3 \\
1 & - 3 & 3 & 5 \\
1 & 4 & 2 & 2
\end{array} \right)$$
The range space of T is denoted by $V$.\\
(i) Determine the dimension of $V$.\\
(ii) Show that the vectors $\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 2 \\ - 1 \\ - 3 \\ 4 \end{array} \right) , \left( \begin{array} { l } 3 \\ 2 \\ 3 \\ 2 \end{array} \right)$ are a basis of $V$.
The set of elements of $\mathbb { R } ^ { 4 }$ which do not belong to $V$ is denoted by $W$.\\
(iii) State, with a reason, whether $W$ is a vector space.\\
(iv) Show that if the vector $\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)$ belongs to $W$ then $x + y \neq z + t$.
\hfill \mbox{\textit{CAIE FP1 2015 Q11 EITHER}}