AI Tutor
Back to The Gram-Schmidt orthogonalization procedure

Exercise 11.3.5

Avg difficulty 3.0

Let

A=[3200251001310013],A=\left[\begin{array}{cccc}3&2&0&0\\ 2&5&1&0\\ 0&1&3&1\\ 0&0&1&3\\ \end{array}\right],

and consider the vector space R4\mathbb{R}^{4} with the inner product given by ⟨v,w⟩=vTAw\langle\mathbf{v},\mathbf{w}\rangle=\mathbf{v}^{T}A\mathbf{w}. Let

v1=[1−111],v2=[2002],andv3=[2023],\mathbf{v}_{1}=\left[\begin{array}{c}1\\ -1\\ 1\\ 1\end{array}\right],\quad\mathbf{v}_{2}=\left[\begin{array}{c}2\\ 0\\ 0\\ 2\end{array}\right],\quad\text{and}\quad\mathbf{v}_{3}=\left[\begin{array}{c}2\\ 0\\ 2\\ 3\end{array}\right],

and let W=span⁡{v1,v2,v3}W=\operatorname{span}\left\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\}. Apply the Gram-Schmidt procedure to v1,v2,v3\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3} to find an orthogonal basis {u1,u2,u3}\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\} for WW with respect to the above inner product.

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, The Gram-Schmidt orthogonalization procedure, Exercise 11.3.5

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.