Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 11 / c

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- Solution c

Solution

Expand an arbitrary vector in the orthonormal eigenbasis:

r = r_{-} e_{-} + r_{0} e_{0} + r_{+} e_{+}, A r = - 6 r_{-} e_{-} + 12 r_{+} e_{+} .

Its image has no component along

e_{0}

, so

(A r) \cdot e_{0} = 0 for every r, e_{0} = \frac{1}{6} (1, 2, 1)^{T} \neq = 0.

Alternatively, symmetry gives

(A r) \cdot e_{0} = r \cdot A e_{0} = 0

. This is image-kernel orthogonality for an adjoint, with

A^{T} = A

.

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