Q40E In Exercise 40 through 46, consi... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q40E (page 217)

In Exercise 40 through 46, consider vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}$ in $R^{4}$ ; we are told that ${\overset{鈫�}{v}}_{i}, {\overset{鈫�}{v}}_{j}$ is the entry $a_{i j}$ of matrix A.
role="math" localid="1659603341436" $A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}]$
Find $| | {\overset{鈫�}{v}}_{2} | |$

Short Answer

Expert verified

$| | {\overset{鈫�}{v}}_{2} | | = 3$

Step by step solution

Step by step solutionStep 1: Given:

The given matrix is $A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}]$

Value of ||v→2||

Firstly, we will write out the matrix A to show its formation based on the given vectors $<v_{i}, v_{j}>$ represents the dot product of the vectors $v_{i}$ and $v_{j}$ .

So, the matrix is:

$A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}] = [\begin{matrix} {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2} & {\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{2} & {\overset{鈫�}{v}}_{2} . {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{2} & {\overset{鈫�}{v}}_{3} . {\overset{鈫�}{v}}_{3} \end{matrix}]$

For:

$\begin{array}{rcl} <v_{2}, v_{2}> & = & | | v_{2} | | \\ = & \sqrt{9} \\ = & 3 \end{array}$

$v_{2} = [\begin{matrix} a \\ b \\ c \\ d \end{matrix}]$

By recognizing that $<v_{2}, v_{2}> = a^{2} + b^{2} + c^{2} + d^{2}$ .also, recall that $| | v_{2} | | = \sqrt{a^{2} + b^{2} + c^{2} + d^{2}}$ . Therefore $<v_{2}, v_{2}>$ can be substituted for $a^{2} + b^{2} + c^{2} + d^{2}$ in the previous equation.

Hence the value of $| | {\overset{鈫�}{v}}_{2} | |$ is 3.

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91影视

Short Answer

Step by step solution

Step by step solutionStep 1: Given:

Value of ||v→2||

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91影视

In Exercise 40 through 46, consider vectors v鈫�1,v鈫�2,v鈫�3in R4; we are told that v鈫�i,v鈫�jis the entry aijof matrix A.role="math" localid="1659603341436" A=[35115920112049]Find ||v鈫�2||

Short Answer

Step by step solution

Step by step solutionStep 1: Given:

Value of ||v→2||

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Discrete Mathematics

Calculus

Applied Mathematics

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Study anywhere. Anytime. Across all devices.