Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 8 / c / ii

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- Solution ii

Solution

Choose

v_{4} = e^{t} (1 - t)

, and denote the new ordered basis by

C = (b_{1}, b_{2}, b_{3}, b_{4})

. The change of basis matrix, with columns expressed in

B

, is

P = 1 - 1 00 11000011 001 - 1, det P = - 4 \neq = 0.

This nonzero determinant proves that the proposed fourth function completes a basis. Express the derivatives in that basis:

F (b_{1}) F (b_{2}) F (b_{3}) F (b_{4}) = - 1 = - \frac{1}{2} b_{1} - \frac{1}{2} b_{2}, = 1 = \frac{1}{2} b_{1} + \frac{1}{2} b_{2}, = e^{t} (2 + t) = \frac{3}{2} b_{3} + \frac{1}{2} b_{4}, = - t e^{t} = - \frac{1}{2} b_{3} + \frac{1}{2} b_{4} .

Reading off columns therefore gives

[F]_{C} = P^{- 1} MP = - \frac{1}{2} - \frac{1}{2} 00 \frac{1}{2} \frac{1}{2} 00 00 \frac{3}{2} \frac{1}{2} 00 - \frac{1}{2} \frac{1}{2} .

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