Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 124 / 3 / i

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- Solution i

Solution

Take the three rank-one quadratic forms

P_{1} = a c, P_{2} = b d, P_{3} = (a + b) (c + d) .

The first two already give

a c

and

b d

, while

a d + b c = P_{3} - P_{1} - P_{2} .

Hence all three required polynomials lie in their linear span.

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