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Exercise 11.3.4

Avg difficulty 3.0

Let

A=[3−10−152023],A=\left[\begin{array}{ccc}3&-1&0\\ -1&5&2\\ 0&2&3\\ \end{array}\right],

and consider the vector space R3\mathbb{R}^{3} with the inner product given by ⟨v,w⟩=vTAw\langle\mathbf{v},\mathbf{w}\rangle=\mathbf{v}^{T}A\mathbf{w}. Let

v1=[102],v2=[−11−5],andv3=[223].\mathbf{v}_{1}=\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right],\quad\mathbf{v}_{2}=\left[\begin{array}{c}-1\\ 1\\ -5\end{array}\right],\quad\text{and}\quad\mathbf{v}_{3}=\left[\begin{array}{c}2\\ 2\\ 3\end{array}\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 R3\mathbb{R}^{3} with respect to the above inner product.

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, The Gram-Schmidt orthogonalization procedure, Exercise 11.3.4

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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.