Linear Algebra With Applications (classic Version)
Linear Algebra With Applications (classic Version)
5th Edition
ISBN: 9780135162972
Author: BRETSCHER, OTTO
Publisher: Pearson Education, Inc.,
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Linear Algebra
In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…
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Chapter 2.3, Problem 34E

For the matrices A in Exercises 33 through 42, compute A 2 = A A , A 3 = A A A , and A 4 . Describe the pattern that emerges, and use this pattern to find A 1001 . Interpret your answers geometrically, in terms of rotations, reflections, shears, and orthogonal projections.

34. [ 1 1 0 1 ]

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Chapter 2 Solutions

Linear Algebra With Applications (classic Version)

Ch. 2.1 - GOAL Use the concept of a linear transformation in...Ch. 2.1 - GOAL Use the concept of a linear transformation in...Ch. 2.1 - GOAL Use the concept of a linear transformation in...Ch. 2.1 - Find the matrix of the linear transformation...Ch. 2.1 - Consider the linear transformation T from 3 to 2...Ch. 2.1 - Consider the transformationT from 2 to 3 given by...Ch. 2.1 - Suppose v1,v2...,vm are arbitrary vectors in n...Ch. 2.1 - Find the inverse of the linear transformation...Ch. 2.1 - In Exercises 9 through 12, decide whether the...Ch. 2.1 - In Exercises 9 through 12, decide whether the...
Ch. 2.1 - In Exercises 9 through 12, decide whether the...Ch. 2.1 - In Exercises 9 through 12, decide whether the...Ch. 2.1 - Prove the following facts: a. The 22 matrix...Ch. 2.1 - a. For which values of the constantk is the matrix...Ch. 2.1 - For which values of the constants a and b is the...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Give a geometric interpretation of the linear...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - Consider the circular face in the accompanying...Ch. 2.1 - In Chapter 1, we mentioned that an old German...Ch. 2.1 - Find an nn matrix A such that Ax=3x , for all x in...Ch. 2.1 - Consider the transformation T from 2 to 2...Ch. 2.1 - Consider the transformation T from 2 to 2 that...Ch. 2.1 - In the example about the French coast guard in...Ch. 2.1 - Let T be a linear transformation from 2 to 2 . Let...Ch. 2.1 - Consider a linear transformation T from 2 to 2 ....Ch. 2.1 - The two column vectors v1 and v2 of a 22 matrix A...Ch. 2.1 - Show that if T is a linear transformation from m...Ch. 2.1 - Prob. 40ECh. 2.1 - Prob. 41ECh. 2.1 - When you represent a three-dimensional object...Ch. 2.1 - a. Consider the vector v=[234] . Is the...Ch. 2.1 - The cross product of two vectors in 3 is given by...Ch. 2.1 - Prob. 45ECh. 2.1 - Prob. 46ECh. 2.1 - Prob. 47ECh. 2.1 - Prob. 48ECh. 2.1 - Prove that if A is a transition matrix and x is a...Ch. 2.1 - Prob. 50ECh. 2.1 - Prob. 51ECh. 2.1 - Prob. 52ECh. 2.1 - Prob. 53ECh. 2.1 - Prob. 54ECh. 2.1 - Prob. 55ECh. 2.1 - For each of the, mini-Webs in Exercises 54 through...Ch. 2.1 - Some parking meters in downtown Geneva,...Ch. 2.1 - Prob. 58ECh. 2.1 - Prob. 59ECh. 2.1 - In the financial pages of a newspaper, one can...Ch. 2.1 - Prob. 61ECh. 2.1 - Prob. 62ECh. 2.1 - Prob. 63ECh. 2.1 - Prob. 64ECh. 2.2 - Sketch the image of the standard L under the...Ch. 2.2 - Find the matrix of a rotation through an angle of...Ch. 2.2 - Consider a linear transformation T from 2 to 3 ....Ch. 2.2 - Interpret the following linear transformation...Ch. 2.2 - The matrix [0.80.60.60.8] represents a rotation....Ch. 2.2 - Let L be the line in 3 that consists of all scalar...Ch. 2.2 - Let L be the line in 3 that consists of all scalar...Ch. 2.2 - Interpret the following linear transformation...Ch. 2.2 - Interpret the following linear transformation...Ch. 2.2 - Find the matrix of the orthogonal projection onto...Ch. 2.2 - Refer to Exercise 10. Find the matrix of the...Ch. 2.2 - Consider a reflection matrix A and a vector x in 2...Ch. 2.2 - Suppose a line L in 2 contains the Unit vector...Ch. 2.2 - Suppose a line L in 3 contains the unit vector...Ch. 2.2 - Suppose a line L in 3 contains the unit vector...Ch. 2.2 - Let T(x)=refL(x) be the reflection about the line...Ch. 2.2 - Consider a matrix A of the form A=[abba] , where...Ch. 2.2 - The linear transformation T(x)=[0.60.80.80.6]x is...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Find the matrices of the linear transformations...Ch. 2.2 - Rotations and reflections have two remarkable...Ch. 2.2 - Find the inverse of the matrix [1k01] ,where k is...Ch. 2.2 - a. Find the scaling matrix A that transforms [21]...Ch. 2.2 - Prob. 27ECh. 2.2 - Prob. 28ECh. 2.2 - Prob. 29ECh. 2.2 - Find a nonzero 22 matrix A such that Ax is...Ch. 2.2 - Prob. 31ECh. 2.2 - Consider the rotation matrix D=[cossinsincos] and...Ch. 2.2 - Consider two nonparallel lines L1 and L2 in 2...Ch. 2.2 - One of the five given matrices represents an...Ch. 2.2 - Prob. 35ECh. 2.2 - Prob. 36ECh. 2.2 - Prob. 37ECh. 2.2 - The determinant of a matrix [abcd] is adbc (wehave...Ch. 2.2 - Describe each of the linear transformations...Ch. 2.2 - Prob. 40ECh. 2.2 - Prob. 41ECh. 2.2 - Prob. 42ECh. 2.2 - Prob. 43ECh. 2.2 - A nonzero matrix of the form A=[abba] represents a...Ch. 2.2 - Prob. 45ECh. 2.2 - A nonzero matrix of the form A=[abba] represents a...Ch. 2.2 - Prob. 47ECh. 2.2 - Prob. 48ECh. 2.2 - Prob. 49ECh. 2.2 - Prob. 50ECh. 2.2 - Prob. 51ECh. 2.2 - Prob. 52ECh. 2.2 - Sketch the image of the unit circle under the...Ch. 2.2 - Prob. 54ECh. 2.2 - Prob. 55ECh. 2.2 - Consider an invertible linear transformation T...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - If possible, compute the matrix products in...Ch. 2.3 - For the matrices A=[ 1 1 1 1],B=[ 1 2 3],C=[ 1 0 1...Ch. 2.3 - Prob. 15ECh. 2.3 - Prob. 16ECh. 2.3 - In the Exercises 17 through 26,find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - In the Exercises 17 through 26, find all matrices...Ch. 2.3 - Prove the distributive laws for matrices:...Ch. 2.3 - Consider an np matrix A, a pm in matrix B, and...Ch. 2.3 - Consider the matrix D=[cossinsincos] . We know...Ch. 2.3 - Consider the lines P and Q in 2 in the...Ch. 2.3 - Consider two matrices A and B whose product ABis...Ch. 2.3 - Prob. 32ECh. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - For the matrices A in Exercises 33 through 42,...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 43 through 48, find a 22matrix A with...Ch. 2.3 - In Exercises 49 through 54, consider the matrices...Ch. 2.3 - In Exercises 49 through 54, consider the matrices...Ch. 2.3 - In Exercises 49 through 54, consider the matrices...Ch. 2.3 - Prob. 52ECh. 2.3 - Prob. 53ECh. 2.3 - Prob. 54ECh. 2.3 - In Exercises 55 through 64,find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - In Exercises 55 through 64, find all matrices X...Ch. 2.3 - Find all upper triangular 22 matrices X such that...Ch. 2.3 - Find all lower triangular 33 matrices X such that...Ch. 2.3 - Prob. 67ECh. 2.3 - Prob. 68ECh. 2.3 - Consider the matrix A2 in Example 4 of Section...Ch. 2.3 - a. Compute A3 for the matrix A in Example 2.3.4....Ch. 2.3 - For the mini-Web in Example 2.3.4, find pages i...Ch. 2.3 - Prob. 72ECh. 2.3 - Prob. 73ECh. 2.3 - Prob. 74ECh. 2.3 - Prob. 75ECh. 2.3 - Prob. 76ECh. 2.3 - Prob. 77ECh. 2.3 - Prob. 78ECh. 2.3 - Prob. 79ECh. 2.3 - Prob. 80ECh. 2.3 - Prob. 81ECh. 2.3 - Prob. 82ECh. 2.3 - Prob. 83ECh. 2.3 - Prob. 84ECh. 2.3 - Prob. 85ECh. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Decide whether the matrices in Exercises 1 through...Ch. 2.4 - Prob. 16ECh. 2.4 - Prob. 17ECh. 2.4 - Prob. 18ECh. 2.4 - Decide whether the linear transformations in...Ch. 2.4 - Decide whether the linear transformations in...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the functions f from to in Exercises 21...Ch. 2.4 - Which of the (nonlinear) tranformtions from 2to...Ch. 2.4 - Which of the (nonlinear) tranformtions from 2to...Ch. 2.4 - Which of the (nonlinear) tranformtions from 2to...Ch. 2.4 - Find the inverse of the linear transformation...Ch. 2.4 - For which values of the constant k is the...Ch. 2.4 - For which values of the constants h and c is the...Ch. 2.4 - For which values of the constants a, b, and c is...Ch. 2.4 - Find all matrices [abcd] such that adbc=1 and A1=A...Ch. 2.4 - Consider the matrices of the form A=[abba] ,where...Ch. 2.4 - Consider the diagonal matrix A=[a000b000c] . a....Ch. 2.4 - Consider the upper triangular 33 matrix...Ch. 2.4 - To determine whether a square matrix A is...Ch. 2.4 - If A is an invertible matrix and c is a nonzero...Ch. 2.4 - Find A1 for A=[1k01] .Ch. 2.4 - Consider a square matrix that differs from the...Ch. 2.4 - Show that if a square matrix A has two equal...Ch. 2.4 - Which of the following linear transformations T...Ch. 2.4 - A square matrix is called a permutation matrix if...Ch. 2.4 - Consider two invertible nn matrices A and B. Is...Ch. 2.4 - Consider the nn matrix Mn , with n2 , that...Ch. 2.4 - To gauge the complexity of a computational task,...Ch. 2.4 - Consider the linear system Ax=b ,where A is an...Ch. 2.4 - Give an example of a noninvertible function f from...Ch. 2.4 - Consider an invertible linear transformation...Ch. 2.4 - Input-Output Analysis. (This exercise builds on...Ch. 2.4 - This exercise refers to exercise 49a. Consider the...Ch. 2.4 - Prob. 51ECh. 2.4 - Prob. 52ECh. 2.4 - Prob. 53ECh. 2.4 - Prob. 54ECh. 2.4 - Prob. 55ECh. 2.4 - Prob. 56ECh. 2.4 - Prob. 57ECh. 2.4 - Prob. 58ECh. 2.4 - Prob. 59ECh. 2.4 - Prob. 60ECh. 2.4 - Prob. 61ECh. 2.4 - In Exercises 55 through 65, show that the given...Ch. 2.4 - Prob. 63ECh. 2.4 - Prob. 64ECh. 2.4 - Prob. 65ECh. 2.4 - Prob. 66ECh. 2.4 - Prob. 67ECh. 2.4 - For two invertible nnmatrices A and B, determine...Ch. 2.4 - Prob. 69ECh. 2.4 - For two invertible nnmatrices A and B, determine...Ch. 2.4 - Prob. 71ECh. 2.4 - Prob. 72ECh. 2.4 - Prob. 73ECh. 2.4 - Prob. 74ECh. 2.4 - For two invertible nnmatrices A and B, determine...Ch. 2.4 - Find all linear transformations T from 2 to 2...Ch. 2.4 - Prob. 77ECh. 2.4 - Prob. 78ECh. 2.4 - Prob. 79ECh. 2.4 - Consider the regular tetrahedron sketched below,...Ch. 2.4 - Find the matrices of the transformations T and L...Ch. 2.4 - Consider the matrix E=[100310001] and an arbitrary...Ch. 2.4 - Are elementary matrices invertible? If so, is the...Ch. 2.4 - a. Justify the following: If A is an nm in matrix,...Ch. 2.4 - a. Justify the following: If A is an nm...Ch. 2.4 - a. Justify the following: Any invertible matrix is...Ch. 2.4 - Write all possible forms of elementary...Ch. 2.4 - Prob. 88ECh. 2.4 - Prob. 89ECh. 2.4 - Prob. 90ECh. 2.4 - Prob. 91ECh. 2.4 - Show that the matrix A=[0110] cannot be written...Ch. 2.4 - In this exercise we will examine which invertible...Ch. 2.4 - Prob. 94ECh. 2.4 - Prob. 95ECh. 2.4 - Prob. 96ECh. 2.4 - Prob. 97ECh. 2.4 - Prob. 98ECh. 2.4 - Prob. 99ECh. 2.4 - Prob. 100ECh. 2.4 - Prob. 101ECh. 2.4 - Prob. 102ECh. 2.4 - Prob. 103ECh. 2.4 - The color of light can be represented in a vector...Ch. 2.4 - Prob. 105ECh. 2.4 - Prob. 106ECh. 2.4 - Prob. 107ECh. 2.4 - Prob. 108ECh. 2 - The matrix [5665] represents a rotation...Ch. 2 - If A is any invertible nn matrix, then A...Ch. 2 - Prob. 3ECh. 2 - Matrix [1/21/21/21/2] represents a rotation.Ch. 2 - Prob. 5ECh. 2 - Prob. 6ECh. 2 - Prob. 7ECh. 2 - Prob. 8ECh. 2 - Prob. 9ECh. 2 - Prob. 10ECh. 2 - Matrix [k25k6] is invertible for all real numbers...Ch. 2 - There exists a real number k such that the matrix...Ch. 2 - Prob. 13ECh. 2 - Prob. 14ECh. 2 - Prob. 15ECh. 2 - Prob. 16ECh. 2 - Prob. 17ECh. 2 - Prob. 18ECh. 2 - Prob. 19ECh. 2 - Prob. 20ECh. 2 - Prob. 21ECh. 2 - Prob. 22ECh. 2 - Prob. 23ECh. 2 - There exists a matrix A such that [1212]A=[1111] .Ch. 2 - Prob. 25ECh. 2 - Prob. 26ECh. 2 - Prob. 27ECh. 2 - There exists a nonzero upper triangular 22 matrix...Ch. 2 - Prob. 29ECh. 2 - Prob. 30ECh. 2 - Prob. 31ECh. 2 - Prob. 32ECh. 2 - Prob. 33ECh. 2 - If A2 is invertible, then matrix A itself must be...Ch. 2 - Prob. 35ECh. 2 - Prob. 36ECh. 2 - Prob. 37ECh. 2 - Prob. 38ECh. 2 - Prob. 39ECh. 2 - Prob. 40ECh. 2 - Prob. 41ECh. 2 - Prob. 42ECh. 2 - Prob. 43ECh. 2 - Prob. 44ECh. 2 - Prob. 45ECh. 2 - Prob. 46ECh. 2 - Prob. 47ECh. 2 - Prob. 48ECh. 2 - Prob. 49ECh. 2 - Prob. 50ECh. 2 - Prob. 51ECh. 2 - Prob. 52ECh. 2 - Prob. 53ECh. 2 - Prob. 54ECh. 2 - Prob. 55ECh. 2 - Prob. 56ECh. 2 - Prob. 57ECh. 2 - Prob. 58ECh. 2 - Prob. 59ECh. 2 - Prob. 60ECh. 2 - Prob. 61ECh. 2 - For every transition matrix A there exists a...
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