Equality in Cauchy–Schwarz

ProofBeginner⏱ 15 minFoundationscauchy-schwarzinner productequality case

Theory: Linear Algebra for Machine Learning →

Problem

Prove that for $x,y \in \R^d$ ,

|\langle x, y\rangle| = \|x\|_2\|y\|_2

if and only if $x$ and $y$ are linearly dependent. Then deduce that the cosine similarity $\cos(x,y) = \langle x,y\rangle/(\|x\|_2\|y\|_2)$ takes the value $\pm 1$ exactly for parallel and anti-parallel vectors.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

Hints — 0 of 3 revealed

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Related problems

All problems

Problem	Field	Type	Difficulty	Time
Equality in Jensen's inequality	Foundations	Proof	Beginner	15 min
Bounding the spectral norm by row and column sums	Foundations	Proof	Intermediate	30 min
Characterising orthogonal projectors	Foundations	Proof	Intermediate	30 min
From distance preservation to inner-product preservation	High-Dim Probability	Proof	Intermediate	30 min
Implement power iteration and verify the convergence rate run	Foundations	Coding	Intermediate	1 hour