AI Tutor
Back to Projection Into a Subspace

Exercise 11

Avg difficulty 3.0

Show that if M⊆RnM\subseteq\mathbb{R}^{n} is a subspace with orthonormal basis ⟨κ⃗1,…,κ⃗n⟩\langle\vec{\kappa}_{1},\ldots,\vec{\kappa}_{n}\rangle then the orthogonal projection of v⃗\vec{v} into MM is this.

(v⃗∙κ⃗1)⋅κ⃗1+⋯+(v⃗∙κ⃗n)⋅κ⃗n(\vec{v}\mathbin{\smash{\bullet}}\vec{\kappa}_{1})\cdot\vec{\kappa}_{1}+\dots+(\vec{v}\mathbin{\smash{\bullet}}\vec{\kappa}_{n})\cdot\vec{\kappa}_{n}

Hefferon, Linear Algebra, Maps Between Spaces, Projection Into a Subspace, Exercise 11

AI Tutor

Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.