Q32E Consider two vectors v鈫�1聽and ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q32E (page 217)

Consider two vectors ${\overset{鈫�}{v}}_{1}$ and localid="1659542347261" ${\overset{鈫�}{v}}_{2}$ in $R^{n}$ . Form the matrix
$G = [\begin{matrix} {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2} \\ {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{2} \end{matrix}]$
For which choices of localid="1659542539965" ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ is the matrix Ginvertible?

Short Answer

Expert verified

The matrix is invertible if and only if ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}$ are not parallel or not lie in the same plane, i.e. ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ will be linearly independent.

Step by step solution

Step by step solutionStep 1: Given

Consider two vectors ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ in $R^{n}$ to Form the matrix $G = [\begin{matrix} {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2} \\ {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{2} \end{matrix}]$

Use Cauchy-Schwarz inequality

The identity is $| \overset{鈫�}{v} 路 \overset{鈫�}{w} | 猢� | | \overset{鈫�}{v} | | 路 | | \overset{鈫�}{w} | |$ .

Since,

role="math" localid="1659543489932" $\begin{array}{rcl} G & = & | | {\overset{鈫�}{v}}_{1} | |^{2} . | | {\overset{鈫�}{v}}_{2} | |^{2} - ({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2}) ({\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{1}) \\ = & | | {\overset{鈫�}{v}}_{1} | |^{2} . | | {\overset{鈫�}{v}}_{2} | |^{2} - {({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2})}^{2} \end{array}$

By using Cauchy-Schwarz inequality:

${\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2} 猢� | | {\overset{鈫�}{v}}_{1} | | 路 | | {\overset{鈫�}{v}}_{2} | | 鈬� {({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2})}^{2} 猢� | | {\overset{鈫�}{v}}_{1} | |^{2} 路 | | {\overset{鈫�}{v}}_{2} | |^{2}$

So, ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}$ will be linearly independent.

Thus, the determinant will be non-vanishing, i.e., the matrix is invertible if and only if ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}$ are not parallel or not lie in the same plane.

Hence, ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ will be linearly independent. if and only if ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}$ are not parallel or not lie in the same plane.

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91影视

Short Answer

Step by step solution

Step by step solutionStep 1: Given

Use Cauchy-Schwarz inequality

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91影视

Consider two vectors v鈫�1and localid="1659542347261" v鈫�2in Rn. Form the matrixG=[v鈫�1.v鈫�1v鈫�1.v鈫�2v鈫�2.v鈫�1v鈫�2.v鈫�2]For which choices of localid="1659542539965" v鈫�1and v鈫�2is the matrix Ginvertible?

Short Answer

Step by step solution

Step by step solutionStep 1: Given

Use Cauchy-Schwarz inequality

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.