Let $\mathbf{x}$ and $\mathbf{y}$ 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. Under what condition does equality hold in the Cauchy-Schwarz inequality?

Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ be unit vectors in $\mathbb{R}^{3}$. Let

$$S=\mathbf{x} \cdot \mathbf{y}+\mathbf{y} \cdot \mathbf{z}+\mathbf{z} \cdot \mathbf{x}$$

Show that for any fixed, linearly independent vectors $\mathbf{x}$ and $\mathbf{y}$, the minimum of $S$ over $\mathbf{z}$ is attained when $\mathbf{z}=\lambda(\mathbf{x}+\mathbf{y})$ for some $\lambda \in \mathbb{R}$, and that for this value of $\lambda$ we have

(i) $\lambda \leqslant-\frac{1}{2}$ (for any choice of $\mathbf{x}$ and $\left.\mathbf{y}\right)$;

(ii) $\lambda=-1$ and $S=-\frac{3}{2}$ in the case where $\mathbf{x} \cdot \mathbf{y}=\cos \frac{2 \pi}{3}$.