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Expert-verified Found in: Page 331 ### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384 # Question: In Exercises 1 and 2, you may assume that $$\left\{ {{{\bf{u}}_{\bf{1}}},...,{{\bf{u}}_{\bf{4}}}} \right\}$$ is an orthogonal basis for $${\mathbb{R}^{\bf{4}}}$$.2. {{\bf{u}}_{\bf{1}}} = \left[ {\begin{aligned}{\bf{1}}\\{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right], {{\bf{u}}_{\bf{2}}} = \left[ {\begin{aligned}{ - {\bf{2}}}\\{\bf{1}}\\{ - {\bf{1}}}\\{\bf{1}}\end{aligned}} \right], {{\bf{u}}_{\bf{3}}} = \left[ {\begin{aligned}{\bf{1}}\\{\bf{1}}\\{ - {\bf{2}}}\\{ - {\bf{1}}}\end{aligned}} \right], {{\bf{u}}_{\bf{4}}} = \left[ {\begin{aligned}{ - {\bf{1}}}\\{\bf{1}}\\{\bf{1}}\\{ - {\bf{2}}}\end{aligned}} \right], {\bf{x}} = \left[ {\begin{aligned}{\bf{4}}\\{\bf{5}}\\{ - {\bf{3}}}\\{\bf{3}}\end{aligned}} \right]Write v as the sum of two vectors, one in $${\bf{Span}}\left\{ {{{\bf{u}}_1}} \right\}$$ and the other in $${\bf{Span}}\left\{ {{{\bf{u}}_2},{{\bf{u}}_3},{{\bf{u}}_{\bf{4}}}} \right\}$$.

The vector v is given as \left[ {\begin{aligned}2\\4\\2\\2\end{aligned}} \right] + \left[ {\begin{aligned}2\\1\\{ - 5}\\1\end{aligned}} \right].

See the step by step solution

## Step 1: Find the orthogonal projection of v on w

The orthogonal projection of v on w can be calculated as follows:

\begin{aligned}{\rm{\hat v}} = \frac{{{\bf{v}} \cdot {{\bf{u}}_1}}}{{{{\bf{u}}_1} \cdot {{\bf{u}}_1}}}{{\bf{u}}_1}\\ = \frac{{4 \times 1 + 5 \times 2 - 3 \times 1 + 3 \times 1}}{{{1^2} + {2^2} + {1^1} + {1^2}}}\left[ {\begin{aligned}1\\2\\1\\1\end{aligned}} \right]\\ = \frac{{14}}{7}\left[ {\begin{aligned}1\\2\\1\\1\end{aligned}} \right]\\ = 2\left[ {\begin{aligned}1\\2\\1\\1\end{aligned}} \right]\\ = \left[ {\begin{aligned}2\\4\\2\\2\end{aligned}} \right]\end{aligned}

The vector \left[ {\begin{aligned}2\\4\\2\\2\end{aligned}} \right] is in the span of $${{\bf{u}}_1}$$.

## Step 2 Find the orthogonal projection of v on H

The orthogonal vector to the projection on w in H can be calculated as follows:

\begin{aligned}{\bf{v}} - {\bf{\hat v}} = \left[ {\begin{aligned}4\\5\\{ - 3}\\3\end{aligned}} \right] - \left[ {\begin{aligned}2\\4\\2\\2\end{aligned}} \right]\\ = \left[ {\begin{aligned}2\\1\\{ - 5}\\1\end{aligned}} \right]\end{aligned}

The vector \left[ {\begin{aligned}2\\1\\{ - 5}\\1\end{aligned}} \right] is in the span of $$\left\{ {{{\bf{u}}_2},{{\bf{u}}_3},{{\bf{u}}_4}} \right\}$$.

Thus, the vector v can be expressed as \left[ {\begin{aligned}2\\4\\2\\2\end{aligned}} \right] + \left[ {\begin{aligned}2\\1\\{ - 5}\\1\end{aligned}} \right]. ### Want to see more solutions like these? 