# Paper 1, Section II, $6 \mathbf{C}$

Let $\mathbf{a}_{1}, \mathbf{a}_{2}$ and $\mathbf{a}_{3}$ be vectors in $\mathbb{R}^{3}$. Give a definition of the dot product, $\mathbf{a}_{1} \cdot \mathbf{a}_{2}$, the cross product, $\mathbf{a}_{1} \times \mathbf{a}_{2}$, and the triple product, $\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}$. Explain what it means to say that the three vectors are linearly independent.

Let $\mathbf{b}_{1}, \mathbf{b}_{2}$ and $\mathbf{b}_{3}$ be vectors in $\mathbb{R}^{3}$. Let $S$ be a $3 \times 3$ matrix with entries $S_{i j}=\mathbf{a}_{i} \cdot \mathbf{b}_{j}$. Show that

$\left(\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}\right)\left(\mathbf{b}_{1} \cdot \mathbf{b}_{2} \times \mathbf{b}_{3}\right)=\operatorname{det}(S)$

Hence show that $S$ is of maximal rank if and only if the sets of vectors $\left\{\mathbf{a}_{1}, \mathbf{a}_{2}\right.$, $\left.\mathbf{a}_{3}\right\}$ and $\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ are both linearly independent.

Now let $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ and $\left\{\mathbf{d}_{1}, \mathbf{d}_{2}, \ldots, \mathbf{d}_{n}\right\}$ be sets of vectors in $\mathbb{R}^{n}$, and let $T$ be an $n \times n$ matrix with entries $T_{i j}=\mathbf{c}_{i} \cdot \mathbf{d}_{j}$. Is it the case that $T$ is of maximal rank if and only if the sets of vectors $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ and $\left\{\mathbf{d}_{1}, \mathbf{d}_{2}, \ldots, \mathbf{d}_{n}\right\}$ are both linearly independent? Justify your answer with a proof or a counterexample.

Given an integer $n>2$, is it always possible to find a set of vectors $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ in $\mathbb{R}^{n}$ with the property that every pair is linearly independent and that every triple is linearly dependent? Justify your answer.