 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-i/4/c/ii/solution

Put

p = (3 x^{2} - 2) / 2

B = (1, 3/2 x, p)

from iii. The columns of the given matrix give

T (1) = - 2 + \frac{3}{2} x^{2} = p, T (x) = x, T (x^{2}) = - 2 + 2 x^{2} .

Consequently

T (p) = \frac{3 T ( x ^{2} ) - 2 T ( 1 )}{2} = - 1.

The images of

B

are therefore

(p, 3/2 x, - 1)

, again an orthonormal basis. By i, $T$ is an orthogonal transformation. Its matrix in this orthonormal basis is

[T]_{B} = 001010 - 1 00 .

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