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Back to Orthogonal projections and Fourier series

Exercise 11.4.2

Avg difficulty 3.0

Consider R4\mathbb{R}^{4} with the usual dot product. Let

u1=[0013],u2=[1100],u3=[1−13−1],andv=[6−2−55].\mathbf{u}_{1}=\left[\begin{array}{r}0\\ 0\\ 1\\ 3\end{array}\right],\quad\mathbf{u}_{2}=\left[\begin{array}{r}1\\ 1\\ 0\\ 0\end{array}\right],\quad\mathbf{u}_{3}=\left[\begin{array}{r}1\\ -1\\ 3\\ -1\end{array}\right],\quad\text{and}\quad\mathbf{v}=\left[\begin{array}{r}6\\ -2\\ -5\\ 5\end{array}\right].

Note that u1\mathbf{u}_{1}, u2\mathbf{u}_{2}, and u3\mathbf{u}_{3} are orthogonal. Find the best approximation of v\mathbf{v} in span⁡{u1,u2,u3}\operatorname{span}\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\}.

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Orthogonal projections and Fourier series, Exercise 11.4.2

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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.