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

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Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 3 / a / iv / Solution by

Codex  0 2026-10-31

The dimension formula for a sum of subspaces and the zero intersection imply

dim (U + U^{⊥}) = dim U + dim U^{⊥} \geq n .

But

U + U^{⊥}

is a vector subspace of the

n

-dimensional space

V

, so its dimension is at most

n

. Equality follows, and the sum is all of

V

. Since the intersection is zero, it is a direct sum:

V = U \oplus U^{⊥} .

The argument remains valid for a degenerate ambient symmetric bilinear form; it uses the stated intersection hypothesis rather than assuming nondegeneracy everywhere.

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