(i) Consider the map from $\mathbb{R}^{4}$ to $\mathbb{R}^{3}$ represented by the matrix

$$\left(\begin{array}{rrrr} \alpha & 1 & 1 & -1 \\ 2 & -\alpha & 0 & -2 \\ -\alpha & 2 & 1 & 1 \end{array}\right)$$

where $\alpha \in \mathbb{R}$. Find the image and kernel of the map for each value of $\alpha$.

(ii) Show that any linear map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ may be written in the form $f(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}$ for some fixed vector $\mathbf{a} \in \mathbb{R}^{n}$. Show further that $\mathbf{a}$ is uniquely determined by $f$.

It is given that $n=4$ and that the vectors

$$\left(\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right),\left(\begin{array}{r} 2 \\ -1 \\ 0 \\ -2 \end{array}\right),\left(\begin{array}{r} -1 \\ 2 \\ 1 \\ 1 \end{array}\right)$$

lie in the kernel of $f$. Determine the set of possible values of a.