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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 264)

TRUE OR FALSE
If A is a square matrix such that $A^{T} A = A A^{T}$ then $k e r (A) = k e r (A^{T})$ .

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of the symmetric matrix

鈥淢atrix B is orthogonal if and only if $B^{T} = B^{- 1}$ 鈥�.

02

Explanation of the solution

Consider the statement as follows:

鈥淚f A is a square matrix such that $A^{T} A = A A^{T}$ , then $k e r (A) = k e r (A^{T})$ .鈥�

Every orthogonal matrix is invertible.

Matrix A is invertible if $k e r (A) = k e r (A^{T})$ .

A is a square matrix such that $A^{T} A = A A^{T}$ as follows:

$A^{T} A = A A^{T} A^{T} = A$

This implies that A is an orthogonal matrix.

This implies that A is an orthogonal matrix.

Since, every orthogonal matrix is invertible and A is invertible.

$k e r (A) = k e r (A^{T}) = 0$

Hence, the given statement 鈥淚f A is a square matrix such that $A^{T} A = A A^{T}$ , then $k e r (A) = k e r (A^{T})$ is true.

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Most popular questions from this chapter

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.

8. $\overset{鈫�}{u} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}], \overset{鈫�}{v} = [\begin{matrix} 2 \\ - 8 \\ 5 \end{matrix}]$

In Exercises 40 through 46, consider vectors $v {鈨�}_{1}, v {鈨�}_{2}, v {鈨�}_{3}$ in $R^{4}$ ; we are told that $v {鈨�}_{i}, v {鈨�}_{j}$ is the entry $a_{i j}$ of matrix A.

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

Find a nonzero vector $\overset{鈬赌}{v}$ in span $({\overset{鈬赌}{v}}_{2} {\overset{鈬赌}{v}}_{3})$ such that $\overset{鈬赌}{v}$ is orthogonal to ${\overset{鈬赌}{v}}_{3}$ .Express as a linear combination of localid="1659441496004" ${\overset{鈬赌}{v}}_{2}$ and ${\overset{鈬赌}{v}}_{3}$ .

Let Abe the matrix of an orthogonal projection. Find $A^{2}$ in two ways:

a.Geometrically. (Consider what happens when you apply an orthogonal projection twice.)

b.By computation, using the formula given in Theorem 5.3.10

Find the length of each of the vectors $\overset{鈬赌}{v}$ In exercises 1 through 3.

$\overset{鈬赌}{v} = [\begin{matrix} 7 \\ 11 \end{matrix}]$

Is there an orthogonal transformation T from $R^{3}$ to $R^{3}$ such that $T [\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}]$

and $T [\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}]$ ?

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