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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: Q9E (page 233)

If the $n 脳 n$ matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well? $B^{- 1}$ .

Short Answer

Expert verified

The Matrix $B^{- 1}$ is orthogonal.

Step by step solution

01

Definition of Orthogonal transformation

An $n 脳 n$ matrix Ais orthogonal if and only if its columns form an orthonormal basis of ${鈩�}^{n}$ .

02

Verification whether the given matrix is orthogonal.

Given thatA andB are orthogonal matrices.

By theorem:

The inverse $A^{- 1}$ of an orthogonal $n 脳 n$ matrixA is orthogonal.

Therefore, $B^{- 1}$ is an orthogonal matrix.

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Most popular questions from this chapter

Prove Theorem 5.1.8d. ${(V^{鈯�})}^{鈯�} = V$ for any subspaceV of ${鈩�}^{n}$ .

Let n be an even integer.In both parts of this problem,let Vbe the subspace of all vector $\overset{鈫�}{x}$ in $R^{n}$
such that $x_{j + 2} = x_{j} + x_{j + 1} 鈭赌 j = 1, 2, . ., n 鈭� 2 .$ .Consider the basis $\overset{鈫�}{v}, \overset{鈫�}{w}$ of V with

$\overset{鈫�}{a} = [\begin{array}{l} 1 \\ a \\ a^{2} \\ . \\ . \\ . \\ a^{n 鈭� 1} \end{array}], \overset{鈫�}{b} = [\begin{array}{l} 1 \\ b \\ b^{2} \\ . \\ . \\ . \\ b^{n 鈭� 1} \end{array}]$

where $a = \frac{1 + \sqrt{5}}{2}$ and $b = \frac{1 鈭� \sqrt{5}}{2}$

a.Show that $\overset{鈫�}{a} 鈥�$ is orthogonal to $\overset{鈫�}{b}$

b.Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix.

For each pair of vectors $\overset{鈫�}{u}$ and $\overset{鈫�}{v}$ listed in Exercises 7 through 9, determine whether the angle $胃$ between $\overset{鈫�}{u}$ and $\overset{鈫�}{v}$ is acute, obtuse, or right.

9. $\overset{鈫�}{u} = [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], \overset{鈫�}{v} = [\begin{matrix} 3 \\ 4 \\ 5 \\ 3 \end{matrix}]$ .

Find the angle $胃$ between each of the pairs of vectors $\overset{鈫�}{u}$ and in exercises 4 through 6.

5. $\overset{鈫�}{u} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}| \overset{鈫�}{V} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}]$ .

If $\overset{鈬赌}{u}$ is a unit vector in $R^{n}$ and $L = span (\overset{鈬赌}{u})$ , then ${proj}_{L} (\overset{鈬赌}{x}) = (\overset{鈬赌}{x} . \overset{鈬赌}{u}) \overset{鈬赌}{u} .$ for all the vectors $\overset{鈬赌}{x} in R^{n}$ .

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