AI Tutor
Back to The dot product

Exercise 2.6.15

Avg difficulty 3.0

Prove the Cauchy Schwarz inequality in Rn\mathbb{R}^{n} as follows. For u,v\mathbf{u},\mathbf{v} vectors, consider

(u−proj⁡v(u))⋅(u−proj⁡v(u))≥0(\mathbf{u}-\operatorname{proj}_{\mathbf{v}}(\mathbf{u}))\boldsymbol{\cdot}(\mathbf{u}-\operatorname{proj}_{\mathbf{v}}(\mathbf{u}))\geq 0

Simplify using the axioms of the dot product and then put in the formula for the projection. Notice that this expression equals 00 and you get equality in the Cauchy Schwarz inequality if and only if u=proj⁡v(u)\mathbf{u}=\operatorname{proj}_{\mathbf{v}}(\mathbf{u}). What is the geometric meaning of u=proj⁡v(u)\mathbf{u}=\operatorname{proj}_{\mathbf{v}}(\mathbf{u})?

Selinger, Matrix Theory and Linear Algebra, Vectors in Rn, The dot product, Exercise 2.6.15

AI Tutor

Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.