Paper 1, Section I, 1A1 A

Vectors and Matrices | Part IA, 2010

Let AA be the matrix representing a linear map Φ:RnRm\Phi: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} with respect to the bases {b1,,bn}\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\} of Rn\mathbb{R}^{n} and {c1,,cm}\left\{\mathbf{c}_{1}, \ldots, \mathbf{c}_{m}\right\} of Rm\mathbb{R}^{m}, so that Φ(bi)=Ajicj\Phi\left(\mathbf{b}_{i}\right)=A_{j i} \mathbf{c}_{j}. Let {b1,,bn}\left\{\mathbf{b}_{1}^{\prime}, \ldots, \mathbf{b}_{n}^{\prime}\right\} be another basis of Rn\mathbb{R}^{n} and let {c1,,cm}\left\{\mathbf{c}_{1}^{\prime}, \ldots, \mathbf{c}_{m}^{\prime}\right\} be another basis of Rm\mathbb{R}^{m}. Show that the matrix AA^{\prime} representing Φ\Phi with respect to these new bases satisfies A=C1ABA^{\prime}=C^{-1} A B with matrices BB and CC which should be defined.

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