Skip to main content

Math Algebra Linear Algebra with Applications (2-Download)

a. Consider an n × m matrix A such that A T A = I m . Is it necessarily truethat A A T = I n ? Explain. h. Consider n × n matrixAsuchthat A T A = I n . I s it necessarily truethat A A T = I n ? Explain.

a. Consider an n × m matrix A such that A T A = I m . Is it necessarily truethat A A T = I n ? Explain. h. Consider n × n matrixAsuchthat A T A = I n . I s it necessarily truethat A A T = I n ? Explain.

Linear Algebra with Applications (2-Download)

Linear Algebra with Applications (2-Download)

5th Edition

ISBN: 9780321796974

Author: Otto Bretscher

Publisher: PEARSON

See similar textbooks

Concept explainers

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements ar…

Videos

Textbook Question

Chapter 5.3, Problem 32E

a. Consider an n × m matrix A such that A T A = I m . Is it necessarily truethat A A T = I n ? Explain.
h. Consider n × n matrixAsuchthat A T A = I n . I s it necessarily truethat A A T = I n ? Explain.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

Blurred answer

Chapter 5.3, Problem 31E

Chapter 5.3, Problem 33E

Chapter 5 Solutions

Linear Algebra with Applications (2-Download)

Ch. 5.1 - Find the length of each of the vector vin...Ch. 5.1 - Find the length of each of the vector vin...Ch. 5.1 - Find the length of each of the vector vin...Ch. 5.1 - Find the angle between each of the pairs of...Ch. 5.1 - Find the angle between each of the pairs of...Ch. 5.1 - Find the angle between each of the pairs of...Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - For each pair of vectors u,vlisted in Exercises 7...Ch. 5.1 - For which value(s) the constant k are the vectors...

Ch. 5.1 - Considerthevector u=[131] and v=[100] in n . a....Ch. 5.1 - Give an algebraic proof for thetriangleinequality...Ch. 5.1 - Leg traction. The accompanying figure shows how a...Ch. 5.1 - Leonardo da Vinci and the resolution of forces....Ch. 5.1 - Consider thevector v=[1234] in 4 . Find a basis of...Ch. 5.1 - Consider the vectors...Ch. 5.1 - Find a basis for W , where W=span([1234],[5678]).Ch. 5.1 - Here is an infinite-dimensional version of...Ch. 5.1 - For a line L in 2 , draw a sketch to interpret the...Ch. 5.1 - Refer to Figure 13 of this section. The least-s...Ch. 5.1 - Find scalara, b, c, d, e, f,g such that the...Ch. 5.1 - Consider a basis v1,v2,...,vm of a subspace V of n...Ch. 5.1 - Prove Theorem 5.1 .8d. (V)=V for any subspace V of...Ch. 5.1 - Prob. 24E Ch. 5.1 - a. Consider a vector v in n , and a scalar k. Show...Ch. 5.1 - Find the orthogonal projection of [494949] onto...Ch. 5.1 - Find the orthogonal projection of 9e1 onto the...Ch. 5.1 - Find the orthogonal projection of [1000] onto the...Ch. 5.1 - Prob. 29E Ch. 5.1 - Consider a subspace V of n and a vector x in n...Ch. 5.1 - Considerthe orthonormal vectors u1,u2,...um , in n...Ch. 5.1 - Consider two vectors v1 and v2 in n . Form the...Ch. 5.1 - Among all the vector in n whose components add up...Ch. 5.1 - Among all the unit vectors in n , find the one for...Ch. 5.1 - Among all the unit vectors u=[xyz] in 3 , find...Ch. 5.1 - There are threeexams in your linear algebra class,...Ch. 5.1 - Consider a plane V in 3 with orthonormal basis...Ch. 5.1 - Consider three unit vectors v1,v2 , and v3 in n ....Ch. 5.1 - Can you find a line L in n and a vector x in n...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Prob. 2E Ch. 5.2 - Prob. 3E Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Prob. 7E Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Prob. 9E Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Perform the Gram—Schmidt process on the...Ch. 5.2 - Consider two linearly independent vector v1=[ab]...Ch. 5.2 - Perform the Gram-Schmidt process on the...Ch. 5.2 - Find an orthonormal basis of the plane x1+x2+x3=0 Ch. 5.2 - Find an orthonormal basis of the kernel of the...Ch. 5.2 - Find an orthonormal basis of the kernel of the...Ch. 5.2 - Find an orthonormal basis of the kernel of the...Ch. 5.2 - Consider the matrix M=12[111111111111][235046007]...Ch. 5.2 - Consider the matrix...Ch. 5.2 - Find the QR factorization of A=[030000200004] .Ch. 5.2 - Find an orthonormal basis u1,u2,u3 of 3 such that...Ch. 5.2 - Consider an invertible nn matrix A whose...Ch. 5.2 - Consider an invertible upper triangular nn matrix...Ch. 5.2 - The two column vectors v1 and v2 of a 22 matrix...Ch. 5.2 - Consider a block matrix A=[A1A2] with linearly...Ch. 5.2 - Consider an nm matrix A with rank(A)m . Is it...Ch. 5.2 - Consider an nm matrix A with rank(A)=m . Is...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - IfA andB are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - Consider an nn matrix A, a vector v in m , and...Ch. 5.3 - Consider an nn matrix A. Show that A is an...Ch. 5.3 - Show that an orthogonal transformation L from n to...Ch. 5.3 - Consider a linear transformation L from m to n...Ch. 5.3 - Are the rows of an orthogonal matrix A...Ch. 5.3 - a. Consider an nm matrix A such that ATA=Im . Is...Ch. 5.3 - Find all orthogonal 22 matrices.Ch. 5.3 - Find all orthogonal 33 matrices of theform...Ch. 5.3 - Find an orthogonal transformation T form 3 to 3...Ch. 5.3 - Find an orthogonal matrix of the form [2/31/...Ch. 5.3 - Is there an orthogonal transformation T from 3 to...Ch. 5.3 - a. Give an example of a (nonzero) skew-symmetric...Ch. 5.3 - Consider a line L in n , spanned by a unit vector...Ch. 5.3 - Consider the subspace W of 4 spanned by the vector...Ch. 5.3 - Find the matrix A of the orthogonal projection...Ch. 5.3 - Let A be the matrix of an orthogonal projection....Ch. 5.3 - Consider a unit vector u in 3 . We define the...Ch. 5.3 - Consider an nm matrix A. Find...Ch. 5.3 - For which nm matrices A docs theequation...Ch. 5.3 - Consider a QRfactorizationM=QR . Show that R=QTM .Ch. 5.3 - If A=QR is a QR factorization, what is the...Ch. 5.3 - Consider an invertible nn matrix A. Can you write...Ch. 5.3 - Consider an invertible nn matrix A. Can you write...Ch. 5.3 - a. Find all nn matrices that are both orthogonal...Ch. 5.3 - a. Consider the matrix product Q1=Q2S , where both...Ch. 5.3 - Find a basis of the space V of all symmetric 33...Ch. 5.3 - Find a basis of the space V of all skew-symmetric...Ch. 5.3 - Find the dimension of the space of alt...Ch. 5.3 - Find the dimension of the space of all symmetric...Ch. 5.3 - Is the transformation L(A)=AT from 23 to 32...Ch. 5.3 - Is the transformation L(A)=AT from mn to nm...Ch. 5.3 - Find image and kernel of the linear transformation...Ch. 5.3 - Find theimage and kernel of the linear...Ch. 5.3 - Find the matrix of the linear transformation...Ch. 5.3 - Find the matrix of the lineartransformation...Ch. 5.3 - Consider the matrix A=[111325220] with...Ch. 5.3 - Consider a symmetric invertible nn matrix A...Ch. 5.3 - This exercise shows one way to define the...Ch. 5.3 - Find all orthogonal 22 matrices A such that all...Ch. 5.3 - Find an orthogonal 22 matrix A such that all the...Ch. 5.3 - Consider a subspace V of n with a basis v1,...,vm...Ch. 5.3 - The formula A(ATA)1AT for 11w matrix of an...Ch. 5.3 - In 4 , consider the subspace W spanned by the...Ch. 5.3 - In all parts of this problem, let V be the...Ch. 5.3 - An nn matrix A is said to be a Hankel, matrix...Ch. 5.3 - Consider a vector v in n of theform v=[11a2an1]...Ch. 5.3 - Let n be an even positive integer. In both parts...Ch. 5.3 - For any integer m, we define the Fibonacci number...Ch. 5.4 - Consider the subspaceim(A) of 2 , where A=[2436] ....Ch. 5.4 - Consider the subspace im(A) of 3 , where...Ch. 5.4 - Considerasubspace V of n . Let v1,...,vp be a...Ch. 5.4 - Let A bean nm matrix. Is the formula ker(A)=im(AT)...Ch. 5.4 - Let V be the solution space of the linear system...Ch. 5.4 - If A is an nm matrix, is the formula im(A)=im(AAT)...Ch. 5.4 - Consider a symmetric nn matrix A. What is the...Ch. 5.4 - Consider a linear transformation L(x)=Ax from n to...Ch. 5.4 - Consider the linear system Ax=b , where A=[1326]...Ch. 5.4 - Consider a consistent system Ax=b . a. Show that...Ch. 5.4 - Consider a linear transformation L(x)=Ax from n to...Ch. 5.4 - Using Exercise 10 as a guide, define theterm...Ch. 5.4 - Consider a linear transformation L(x)=Ax from n to...Ch. 5.4 - In the accompanying figure, we show the kernel and...Ch. 5.4 - Consider an mn matrix A with ker(A)={0} . Showthat...Ch. 5.4 - Use the formula (imA)=ker(AT) to prove theequation...Ch. 5.4 - Does the equation rank(A)=rank(ATA) hold for all...Ch. 5.4 - Does the equation rank(ATA)=rank(AAT) hold for all...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - By using paper and pencil, find the least-squares...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Consider an inconsistent linear system Ax=b ,...Ch. 5.4 - Consider an orthonormal basis u1,u2,...,un , in n...Ch. 5.4 - Find theleast-squares solution of the system Ax=b...Ch. 5.4 - Fit a linear function of the form f(t)=c0+c1t...Ch. 5.4 - Fit a linear function of the form f(t)=c0+c1t to...Ch. 5.4 - Fit a quadratic polynomial to the data points...Ch. 5.4 - Find the trigonometric function of the form...Ch. 5.4 - Find the function of the form...Ch. 5.4 - Suppose you wish to fit a function of the form...Ch. 5.4 - Let S (t) be the number of daylight hours on the t...Ch. 5.4 - Prob. 37E Ch. 5.4 - In the accompanying table, we list the height h,...Ch. 5.4 - In the accompanying table, we list the estimated...Ch. 5.4 - Prob. 40E Ch. 5.4 - Prob. 41E Ch. 5.4 - Prob. 42E Ch. 5.5 - In C[a,b] , define the product f,g=abf(t)g(t)dt ....Ch. 5.5 - Does the equation f,g+h=f,g+f,h hold for all...Ch. 5.5 - Consider a matrix S in nn . In n , define the...Ch. 5.5 - In nm , consider the inner product A,B=trace(ATB)...Ch. 5.5 - Is A,B=trace(ABT) an inner product in nm ?(The...Ch. 5.5 - a. Consider an nm matrix P and an mn matrixQ. Show...Ch. 5.5 - Prob. 7E Ch. 5.5 - Prob. 8E Ch. 5.5 - Prob. 9E Ch. 5.5 - Consider the space P2 with inner product...Ch. 5.5 - Prob. 11E Ch. 5.5 - Prob. 12E Ch. 5.5 - For a function f in C[,] (with the inner...Ch. 5.5 - Which of the following is an inner product in P2...Ch. 5.5 - Prob. 15E Ch. 5.5 - Prob. 16E Ch. 5.5 - Prob. 17E Ch. 5.5 - Prob. 18E Ch. 5.5 - Prob. 19E Ch. 5.5 - Prob. 20E Ch. 5.5 - Prob. 21E Ch. 5.5 - Prob. 22E Ch. 5.5 - Prob. 23E Ch. 5.5 - Consider the linear space P of all polynomials,...Ch. 5.5 - Prob. 25E Ch. 5.5 - Prob. 26E Ch. 5.5 - Find the Fourier coefficients of the piecewise...Ch. 5.5 - Prob. 28E Ch. 5.5 - Prob. 29E Ch. 5.5 - Prob. 30E Ch. 5.5 - Gaussian integration,In an introductory...Ch. 5.5 - In the space C[1,1] , we introduce the inner...Ch. 5.5 - a. Let w(t) be a positive-valued function in...Ch. 5.5 - In the space C[1,1] , we define the inner product...Ch. 5.5 - In this exercise, we compare the inner products...Ch. 5 - If T is a linear transformation from n to n...Ch. 5 - If A is an invertible matrix, then the equation...Ch. 5 - Prob. 3E Ch. 5 - Prob. 4E Ch. 5 - Prob. 5E Ch. 5 - Prob. 6E Ch. 5 - All nonzero symmetric matrices are invertible.Ch. 5 - Prob. 8E Ch. 5 - If u is a unit vector in n , and L=span(u) , then...Ch. 5 - Prob. 10E Ch. 5 - Prob. 11E Ch. 5 - Prob. 12E Ch. 5 - If matrix A is orthogonal, then AT must be...Ch. 5 - If A and B are symmetric nn matrices, then AB...Ch. 5 - Prob. 15E Ch. 5 - If A is any matrix with ker(A)={0} , then the...Ch. 5 - If A and B are symmetric nn matrices, then...Ch. 5 - Prob. 18E Ch. 5 - Prob. 19E Ch. 5 - Prob. 20E Ch. 5 - Prob. 21E Ch. 5 - Prob. 22E Ch. 5 - Prob. 23E Ch. 5 - Prob. 24E Ch. 5 - Prob. 25E Ch. 5 - Prob. 26E Ch. 5 - Prob. 27E Ch. 5 - If A is a symmetric matrix, vector v is in the...Ch. 5 - The formula ker(A)=ker(ATA) holds for all matrices...Ch. 5 - Prob. 30E Ch. 5 - Prob. 31E Ch. 5 - Prob. 32E Ch. 5 - If A is an invertible matrix such that A1=A , then...Ch. 5 - Prob. 34E Ch. 5 - The formula (kerB)=im(BT) holds for all matrices...Ch. 5 - The matrix ATA is symmetric for all matrices A.Ch. 5 - If matrix A is similar to B and A is orthogonal,...Ch. 5 - Prob. 38E Ch. 5 - If matrix A is symmetric and matrix S is...Ch. 5 - If A is a square matrix such that ATA=AAT , then...Ch. 5 - Any square matrix can be written as the sum of a...Ch. 5 - If x1,x2,...,xn are any real numbers, then...Ch. 5 - If AAT=A2 for a 22 matrix A, then A must...Ch. 5 - If V is a subspace of n and x is a vector in n ,...Ch. 5 - If A is an nn matrix such that Au=1 for all...Ch. 5 - If A is any symmetric 22 matrix, then there must...Ch. 5 - There exists a basis of 22 that consists of...Ch. 5 - If A=[1221] , then the matrix Q in the QR...Ch. 5 - There exists a linear transformation L from 33 to...Ch. 5 - If a 33 matrix A represents the orthogonal...

Additional Math Textbook Solutions

Find more solutions based on key concepts

the given equation.

Algebra: Structure And Method, Book 1

In Exercises 1-6, consider the numbers 23, 6, 3, 2.45, 266, 18.4, 11, 273, 516, 7.151551555 , 35, 35,87, 0, 16....

College Algebra: Graphs and Models (6th Edition)

True or False? Explain. 1. Any set of ordered pairs is a function.

College Algebra (6th Edition)

Complete each statement with the correct term from the column on the right. Some of the choices may not be used...

Intermediate Algebra (12th Edition)

In each of Exercises 21–30, draw a linear graph to represent the given information. Be sure to label and number...

Elementary Algebra: Concepts and Applications (10th Edition)

Skill Practice a. Write the set represented by the graph in interval notation and set-builder notation. b. Give...

College Algebra (Collegiate Math)

Knowledge Booster

Background pattern image

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Similar questions

Which of the following operations can we perform for a matrix A of any dimension? (i) A+A (ii) 2A (iii) AA
Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Write the third column of the matrix as a linear combination of the first two columns if possible. [102422751]

Recommended textbooks for you

Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning

Text book image

Linear Algebra: A Modern Introduction

Algebra

ISBN:9781285463247

Author:David Poole

Publisher:Cengage Learning

Text book image

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:9781305658004

Author:Ron Larson

Publisher:Cengage Learning

Text book image

College Algebra (MindTap Course List)

Algebra

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:Cengage Learning

Text book image

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:9781133382119

Author:Swokowski

Publisher:Cengage

Text book image

College Algebra

Algebra

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

Text book image

Algebra and Trigonometry (MindTap Course List)

Algebra

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

SEE MORE TEXTBOOKS

Matrix Operations Full Length; Author: ProfRobBob;https://www.youtube.com/watch?v=K5BLNZw7UeU;License: Standard YouTube License, CC-BY

Intro to Matrices; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=yRwQ7A6jVLk;License: Standard YouTube License, CC-BY