 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2014/3/b/i/solution

Use the

m

-dimensional space

V_{m}

for this change of basis. With the columns of

P

recording the new basis in the old one, bilinearity gives

φ (f_{i}, f_{j}) = φ (k = 1 \sum m P_{ki} e_{k}, ℓ = 1 \sum m P_{ℓ j} e_{ℓ}) = k, ℓ = 1 \sum m P_{ki} (A_{m})_{k ℓ} P_{ℓ j} = (P^{T} A_{m} P)_{ij} .

Thus the matrices of the bilinear form satisfy

B_{m} = P^{T} A_{m} P .

This is matrix congruence, rather than the similarity rule for an endomorphism: both arguments of the bilinear form change basis.

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