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Exercise 11.4.1

Avg difficulty 3.0

Consider R3\mathbb{R}^{3} with the usual dot product. Let

u1=[−111],u2=[−1−21],andv=[−153].\mathbf{u}_{1}=\left[\begin{array}{r}-1\\ 1\\ 1\end{array}\right],\quad\mathbf{u}_{2}=\left[\begin{array}{r}-1\\ -2\\ 1\end{array}\right],\quad\text{and}\quad\mathbf{v}=\left[\begin{array}{r}-1\\ 5\\ 3\end{array}\right].

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

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Orthogonal projections and Fourier series, Exercise 11.4.1

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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.