Linear Algebra with Applications (2-Download)
Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
bartleby

Videos

Textbook Question
Book Icon
Chapter 2.1, Problem 1E

GOAL Use the concept of a linear transformation in terms of the formula y → = A x → , and interpret simple linear transformations geometrically Find the inverse of a linear transformation from ℝ 2 to ℝ 2 (if it exists). Find the matrix of a linear transformation column by column.

Consider the transformations from ℝ 3 to ℝ 3 defined in Exercises 1 through 3. Which of these transformations are linear?

1. y 1 = 2 x 2 y 2 = x 2 + 2 y 3 = 2 x 2

Expert Solution & Answer
Check Mark
bartleby

Learn your wayIncludes step-by-step video

See solution
Check out sample textbook solution
Blurred answer
01:18
Previous
Chapter 1, Problem 50E
Next
Chapter 2.1, Problem 2E
Students have asked these similar questions
Topic is Linear Transformation: Given Function:T: R2 -> R2 , T(x,y) = (x-2y , 2y-x). Provide the standard matrix and prove that it is a linear transformation
answer true if the statementis always true and false otherwise. In the case ofa true statement, explain or prove your answer. In thecase of a false statement, give an example to show thatthe statement is not always true. Let L: Rn → Rn be a linear transformation. If L (x1) = L (x2), then the vectors x1 and x2 must be equal.
answer true if the statementis always true and false otherwise. In the case ofa true statement, explain or prove your answer. In thecase of a false statement, give an example to show thatthe statement is not always true. Let L: R2 → R2 be a linear transformation, and let A be the standard matrix representation of L. If L2 is defined by L2(x) = L (L (x)) for all x ∈ R2 then L2 is a linear transformation and its standard matrix representation is A2.

Chapter 2 Solutions

Linear Algebra with Applications (2-Download)

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...

Additional Math Textbook Solutions

Find more solutions based on key concepts
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)

The numbers in the set { x|x = a b , where a, b are integers and b ≠ 0 } are called ---------- numbers.

College Algebra (10th Edition)

Solve each formula for the given letter . [2.3] What percent of 60 is 42? [2.4]

Elementary and Intermediate Algebra: Concepts and Applications (7th Edition)

In Exercises 1 and 2, determine if the function f defined by the arrow diagram has an inverse. If it does, crea...

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)

A faucet can fill a sink in 5 minutes. It takes twice as long for the drain to empty the sink. How long will it...

Intermediate Algebra for College Students (7th Edition)

NOTE: Write your answers using interval notation when appropriate. CHECKING ANALYTIC SKILLS Fill in each blank ...

A Graphical Approach to College Algebra (6th Edition)

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
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
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
Text book image
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
SEE MORE TEXTBOOKS
Linear Transformations on Vector Spaces; Author: Professor Dave Explains;https://www.youtube.com/watch?v=is1cg5yhdds;License: Standard YouTube License, CC-BY
Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY