Let $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ and $\mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right)$ be non-zero vectors in $\mathbb{R}^{n}$. What is meant by saying that $\mathbf{x}$ and $\mathbf{y}$ are linearly independent? What is the dimension of the subspace of $\mathbb{R}^{n}$ spanned by $\mathbf{x}$ and $\mathbf{y}$ if they are (1) linearly independent, (2) linearly dependent?

Define the scalar product $\mathbf{x} \cdot \mathbf{y}$ for $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$. Define the corresponding norm $\|\mathbf{x}\|$ of $\mathbf{x} \in \mathbb{R}^{n}$. State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality.

By means of a sketch, give a geometric interpretation of the scalar product $\mathbf{x} \cdot \mathbf{y}$ in the case $n=3$, relating the value of $\mathbf{x} \cdot \mathbf{y}$ to the angle $\alpha$ between the directions of $\mathbf{x}$ and $\mathbf{y}$.

What is a unit vector? Let $\mathbf{u}, \mathbf{v}, \mathbf{w}$ be unit vectors in $\mathbb{R}^{3}$. Let

$$S=\mathbf{u} \cdot \mathbf{v}+\mathbf{v} \cdot \mathbf{w}+\mathbf{w} \cdot \mathbf{u}$$

Show that

(i) for any fixed, linearly independent $\mathbf{u}$ and $\mathbf{v}$, the minimum of $S$ over $\mathbf{w}$ is attained when $\mathbf{w}=\lambda(\mathbf{u}+\mathbf{v})$ for some $\lambda \in \mathbb{R}$;

(ii) $\lambda \leqslant-\frac{1}{2}$ in all cases;

(iii) $\lambda=-1$ and $S=-3 / 2$ in the case where $\mathbf{u} \cdot \mathbf{v}=\cos (2 \pi / 3)$.