Paper 1, Section II, $\mathbf{6 B}$

Let $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ be the standard basis vectors of $\mathbb{R}^{3}$. A second set of vectors $\mathbf{f}_{1}, \mathbf{f}_{2}, \mathbf{f}_{3}$ are defined with respect to the standard basis by

$\mathbf{f}_{j}=\sum_{i=1}^{3} P_{i j} \mathbf{e}_{i}, \quad j=1,2,3$

The $P_{i j}$ are the elements of the $3 \times 3$ matrix $P$. State the condition on $P$ under which the set $\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \mathbf{f}_{3}\right\}$ forms a basis of $\mathbb{R}^{3}$.

Define the matrix $A$ that, for a given linear transformation $\alpha$, gives the relation between the components of any vector $\mathbf{v}$ and those of the corresponding $\alpha(\mathbf{v})$, with the components specified with respect to the standard basis.

Show that the relation between the matrix $A$ and the matrix $\tilde{A}$ of the same transformation with respect to the second basis $\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \mathbf{f}_{3}\right\}$ is

$\tilde{A}=P^{-1} A P$

Consider the matrix

$A=\left(\begin{array}{ccc} 2 & 6 & 2 \\ 0 & -1 & -1 \\ 0 & 6 & 4 \end{array}\right)$

Find a matrix $P$ such that $B=P^{-1} A P$ is diagonal. Give the elements of $B$ and demonstrate explicitly that the relation between $A$ and $B$ holds.

Give the elements of $A^{n} P$ for any positive integer $n$.

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