Paper 1, Section II, 7C7 \mathrm{C}

Vectors and Matrices | Part IA, 2014

Let A:C2C2\mathcal{A}: \mathbb{C}^{2} \rightarrow \mathbb{C}^{2} be the linear map

A(zw)=(zeiθ+wweiϕ+z)\mathcal{A}\left(\begin{array}{c} z \\ w \end{array}\right)=\left(\begin{array}{c} z \mathrm{e}^{i \theta}+w \\ w \mathrm{e}^{-i \phi}+z \end{array}\right)

where θ\theta and ϕ\phi are real constants. Write down the matrix AA of A\mathcal{A} with respect to the standard basis of C2\mathbb{C}^{2} and show that detA=2isin12(θϕ)exp(12i(θϕ))\operatorname{det} A=2 i \sin \frac{1}{2}(\theta-\phi) \exp \left(\frac{1}{2} i(\theta-\phi)\right).

Let R:C2R4\mathcal{R}: \mathbb{C}^{2} \rightarrow \mathbb{R}^{4} be the invertible map

R(zw)=(RezImzRewImw)\mathcal{R}\left(\begin{array}{c} z \\ w \end{array}\right)=\left(\begin{array}{l} \operatorname{Re} z \\ \operatorname{Im} z \\ \operatorname{Re} w \\ \operatorname{Im} w \end{array}\right)

and define a linear map B:R4R4\mathcal{B}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4} by B=RAR1\mathcal{B}=\mathcal{R} \mathcal{A} \mathcal{R}^{-1}. Find the image of each of the standard basis vectors of R4\mathbb{R}^{4} under both R1\mathcal{R}^{-1} and B\mathcal{B}. Hence, or otherwise, find the matrix BB of B\mathcal{B} with respect to the standard basis of R4\mathbb{R}^{4} and verify that detB=detA2\operatorname{det} B=|\operatorname{det} A|^{2}.

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