Algebra and Geometry | Part IA, 2003

The mapping α\alpha of R3\mathbb{R}^{3} into itself is a reflection in the plane x2=x3x_{2}=x_{3}. Find the matrix AA of α\alpha with respect to any basis of your choice, which should be specified.

The mapping β\beta of R3\mathbb{R}^{3} into itself is a rotation about the line x1=x2=x3x_{1}=x_{2}=x_{3} through 2π/32 \pi / 3, followed by a dilatation by a factor of 2 . Find the matrix BB of β\beta with respect to a choice of basis that should again be specified.

Show explicitly that

B3=8A2B^{3}=8 A^{2}

and explain why this must hold, irrespective of your choices of bases.

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