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

Vectors and Matrices | Part IA, 2019

Let e1,e2,e3\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3} be the standard basis vectors of R3\mathbb{R}^{3}. A second set of vectors f1,f2,f3\mathbf{f}_{1}, \mathbf{f}_{2}, \mathbf{f}_{3} are defined with respect to the standard basis by

fj=i=13Pijei,j=1,2,3\mathbf{f}_{j}=\sum_{i=1}^{3} P_{i j} \mathbf{e}_{i}, \quad j=1,2,3

The PijP_{i j} are the elements of the 3×33 \times 3 matrix PP. State the condition on PP under which the set {f1,f2,f3}\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \mathbf{f}_{3}\right\} forms a basis of R3\mathbb{R}^{3}.

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

Show that the relation between the matrix AA and the matrix A~\tilde{A} of the same transformation with respect to the second basis {f1,f2,f3}\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \mathbf{f}_{3}\right\} is

A~=P1AP\tilde{A}=P^{-1} A P

Consider the matrix

A=(262011064)A=\left(\begin{array}{ccc} 2 & 6 & 2 \\ 0 & -1 & -1 \\ 0 & 6 & 4 \end{array}\right)

Find a matrix PP such that B=P1APB=P^{-1} A P is diagonal. Give the elements of BB and demonstrate explicitly that the relation between AA and BB holds.

Give the elements of AnPA^{n} P for any positive integer nn.

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