Paper 1, Section I, A

Vectors and Matrices | Part IA, 2018

The map Φ(x)=α(nx)nn×(n×x)\boldsymbol{\Phi}(\mathbf{x})=\alpha(\mathbf{n} \cdot \mathbf{x}) \mathbf{n}-\mathbf{n} \times(\mathbf{n} \times \mathbf{x}) is defined for xR3\mathbf{x} \in \mathbb{R}^{3}, where n\mathbf{n} is a unit vector in R3\mathbb{R}^{3} and α\alpha is a real constant.

(i) Find the values of α\alpha for which the inverse map Φ1\Phi^{-1} exists, as well as the inverse map itself in these cases.

(ii) When Φ\boldsymbol{\Phi} is not invertible, find its image and kernel. What is the value of the rank and the value of the nullity of Φ\Phi ?

(iii) Let y=Φ(x)\mathbf{y}=\mathbf{\Phi}(\mathbf{x}). Find the components AijA_{i j} of the matrix AA such that yi=Aijxjy_{i}=A_{i j} x_{j}. When Φ\Phi is invertible, find the components of the matrix BB such that xi=Bijyjx_{i}=B_{i j} y_{j}.

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