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ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2005/algebra/2/ii/solution

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2005 / Algebra / 2 / ii / Solution by

Codex  0 2026-10-31

v = 0

, equality is immediate. If

v \neq = 0

, the nonnegative squared norm

0 \leq u - \frac{⟨ u , v ⟩}{∥ v ∥ ^{2}} v^{2} = ∥ u ∥^{2} - \frac{⟨ u , v ⟩ ^{2}}{∥ v ∥ ^{2}}

(1)

proves the Cauchy-Schwarz inequality. Equality holds precisely when the displayed vector vanishes. Thus equality is equivalent to linear dependence of $u, v$ , including the cases with a zero vector.

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