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

If $v=0$, equality is immediate. If $v\ne 0$, the nonnegative squared normproves 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.