The matrix

$$A_{\alpha}=\left(\begin{array}{ccc} 1 & -1 & 2 \alpha+1 \\ 1 & \alpha-1 & 1 \\ 1+\alpha & -1 & \alpha^{2}+4 \alpha+1 \end{array}\right)$$

defines a linear map $\Phi_{\alpha}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by $\Phi_{\alpha}(\mathbf{x})=A_{\alpha} \mathbf{x}$. Find a basis for the kernel of $\Phi_{\alpha}$ for all values of $\alpha \in \mathbb{R}$.

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ and $\mathcal{C}=\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \mathbf{c}_{3}\right\}$ be bases of $\mathbb{R}^{3}$. Show that there exists a matrix $S$, to be determined in terms of $\mathcal{B}$ and $\mathcal{C}$, such that, for every linear mapping $\Phi$, if $\Phi$ has matrix $A$ with respect to $\mathcal{B}$ and matrix $A^{\prime}$ with respect to $\mathcal{C}$, then $A^{\prime}=S^{-1} A S$.

For the bases

$$\mathcal{B}=\left\{\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right)\right\}, \mathcal{C}=\left\{\left(\begin{array}{l} 1 \\ 2 \\ 2 \end{array}\right),\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right),\left(\begin{array}{l} 2 \\ 3 \\ 2 \end{array}\right)\right\},$$

find the basis transformation matrix $S$ and calculate $S^{-1} A_{0} S$.