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Linear Algebra and its Applications
Found in: Page 331
Linear Algebra and its Applications

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

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Short Answer

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 \(\left[ {\begin{aligned}2\\4\\2\\2\end{aligned}} \right] + \left[ {\begin{aligned}2\\1\\{ - 5}\\1\end{aligned}} \right]\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

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 \(\left[ {\begin{aligned}2\\4\\2\\2\end{aligned}} \right] + \left[ {\begin{aligned}2\\1\\{ - 5}\\1\end{aligned}} \right]\).

Most popular questions for Math Textbooks

Given data for a least-squares problem, \(\left( {{x_1},{y_1}} \right), \ldots ,\left( {{x_n},{y_n}} \right)\), the following abbreviations are helpful:

\(\begin{aligned}{l}\sum x = \sum\nolimits_{i = 1}^n {{x_i}} ,{\rm{ }}\sum {{x^2}} = \sum\nolimits_{i = 1}^n {x_i^2} ,\\\sum y = \sum\nolimits_{i = 1}^n {{y_i}} ,{\rm{ }}\sum {xy} = \sum\nolimits_{i = 1}^n {{x_i}{y_i}} \end{aligned}\)

The normal equations for a least-squares line \(y = {\hat \beta _0} + {\hat \beta _1}x\)may be written in the form

\(\begin{aligned}{{\hat \beta }_0} + {{\hat \beta }_1}\sum x = \sum y \\{{\hat \beta }_0}\sum x + {{\hat \beta }_1}\sum {{x^2}} = \sum {xy} {\rm{ (7)}}\end{aligned}\)

16. Use a matrix inverse to solve the system of equations in (7) and thereby obtain formulas for \({\hat \beta _0}\) , and that appear in many statistics texts.

In Exercises 9-12, find a unit vector in the direction of the given vector.

9. \(\left( {\begin{aligned}{*{20}{c}}{ - 30}\\{40}\end{aligned}} \right)\)

In Exercises 9-12, find a unit vector in the direction of the given vector.

10. \(\left( {\begin{aligned}{*{20}{c}}{ - 6}\\4\\{ - 3}\end{aligned}} \right)\)

In exercises 1-6, determine which sets of vectors are orthogonal.

\(\left[ {\begin{align}{ 2}\\{ - 7}\\{-1}\end{align}} \right]\), \(\left[ {\begin{align}{ - 6}\\{ - 3}\\9\end{align}} \right]\), \(\left[ {\begin{align}{ 3}\\{ 1}\\{-1}\end{align}} \right]\)

In exercises 1-6, determine which sets of vectors are orthogonal.

\(\left[ {\begin{align} 2\\{-5}\\{-3}\end{align}} \right]\), \(\left[ {\begin{align}0\\0\\0\end{align}} \right]\), \(\left[ {\begin{align} 4\\{ - 2}\\6\end{align}} \right]\)

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