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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: Q3P (page 214)

Find the length of each of the vectors $\overset{鈬赌}{v}$ In exercises 1 through 3.
3. $\overset{鈬赌}{v} = [\begin{matrix} 2 \\ 3 \\ 4 \\ 5 \end{matrix}]$

Short Answer

Expert verified

The length of the vector is about 7.35.

Step by step solution

01

Consider the definition below

The length(or magnitude or norm) of a vector $\overset{鈬赌}{v} in R^{n} is | | \overset{鈬赌}{v} | | = \sqrt{\overset{鈬赌}{v} . \overset{鈬赌}{v}}$

02

Use the definition

By using the above definition the length of the vector will be.

$| | \overset{鈬赌}{v} | | = \sqrt{\overset{鈬赌}{v} . \overset{鈬赌}{v}} = \sqrt{[2 3 4 5] [\begin{matrix} 2 \\ 3 \\ 4 \\ 5 \end{matrix}]} = \sqrt{2 - 2 + 3 - 3 + 4 - 4 + 5 - 5} = \sqrt{4 + 9 + 16 + 25} = \sqrt{54} 鈮� 7.35$

Hence, the value of the vector is about 7.35.

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Most popular questions from this chapter

If A and B are arbitrary $n 脳 n$ matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

$B (A + A^{螕}) B^{螕}$ .

If the nxnmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well? $A^{T}$ .

Consider an invertible n脳nmatrix A. Can you write Aas A=LQ, where Lis a lowertriangular matrix andQis orthogonal? Hint: Consider the QRfactorizationof $A^{T}$ .

a.Consider a vector $\overset{鈯�}{v}$ in $R^{n}$ , and a scalar k. Show that $| | k \overset{鈫�}{v} | | = | k | . | | \overset{鈫�}{v} | |$

b.Show that if $\overset{鈯�}{v}$ is a nonzero vector in $R^{n}$ , then

$\overset{鈫�}{u} = \frac{1}{| | \overset{鈫�}{v} | |}$ is a unit vector.

If Ais an $n 脳 m$ matrix, is the formula $im (A) = im ({AA}^{T})$ necessarily true? Explain.

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