Linear Algebra With Applications (classic Version)
5th Edition
ISBN: 9780135162972
Author: BRETSCHER, OTTO
Publisher: Pearson Education, Inc.,
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 4.2, Problem 22E
Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms.
22.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Let
2
A =
4
3
-4
0
1
(a) Show that v =
eigenvalue.
()
is an eigenvector of A and find the corresponding
(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)
may be useful.
(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.
(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.
(c) Let
6
0 0
A =
-10 4 8
5 1 2
(i) Find the characteristic polynomial of A and factorise it.
(ii) Determine all eigenvalues of A and find bases for the corresponding
eigenspaces.
(iii) Is A diagonalisable? Give reasons for your answer.
most 2, and let
Let P2 denote the vector space of polynomials of degree at
D: P2➡ P2
be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative
p'(t)
2at+b, that is,
D(p) = p'.
(a) Prove that D is a linear transformation.
(b) Find a basis for the kernel ker(D) of the linear transformation D and compute its
nullity.
(c) Find a basis for the image im(D) of the linear transformation D and compute its
rank.
(d) Verify that the Rank-Nullity Theorem holds for the linear transformation D.
(e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.
Chapter 4 Solutions
Linear Algebra With Applications (classic Version)
Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...
Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Let V be the space of all infinite sequences of...Ch. 4.1 - Let V be the space of all infinite sequences of...Ch. 4.1 - Let V be the space of all infinite sequences of...Ch. 4.1 - Let V be the space of all infinite sequences of...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Prob. 31ECh. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Prob. 33ECh. 4.1 - Find a basis for each of the spaces V in Exercises...Ch. 4.1 - Prob. 35ECh. 4.1 - Prob. 36ECh. 4.1 - Prob. 37ECh. 4.1 - Prob. 38ECh. 4.1 - Prob. 39ECh. 4.1 - If c is any vector in n , what are the possible...Ch. 4.1 - Prob. 41ECh. 4.1 - Prob. 42ECh. 4.1 - Prob. 43ECh. 4.1 - Prob. 44ECh. 4.1 - Prob. 45ECh. 4.1 - In the linear space of infinite sequences,...Ch. 4.1 - A function f(t) from to is called even if...Ch. 4.1 - Prob. 48ECh. 4.1 - Let L(m,n) be the set of all linear...Ch. 4.1 - Prob. 50ECh. 4.1 - Prob. 51ECh. 4.1 - Make up a second-order linear DE whose solution...Ch. 4.1 - Show that in an n-dimensional linear space we can...Ch. 4.1 - Show that if W is a subspace of an n-dimensional...Ch. 4.1 - Show that the space F(,) of all functions from to...Ch. 4.1 - Show that the space of infinite sequences of real...Ch. 4.1 - We say that a linear space V is finitely generated...Ch. 4.1 - In this exercise we will show that the functions...Ch. 4.1 - Show that if 0 is the neutral element of a linear...Ch. 4.1 - Consider the sequence (f0,f1,f2) recursively...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 15ECh. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 17ECh. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 21ECh. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 23ECh. 4.2 - Prob. 24ECh. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 35ECh. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 41ECh. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 46ECh. 4.2 - Find out which of the transformations in Exercises...Ch. 4.2 - Prob. 48ECh. 4.2 - Prob. 49ECh. 4.2 - Prob. 50ECh. 4.2 - Prob. 51ECh. 4.2 - Prob. 52ECh. 4.2 - Find the image, rank, kernel, and nullity of the...Ch. 4.2 - Find the image, rank, kernel, and nullity of the...Ch. 4.2 - Find the image and kernel of the transformation T...Ch. 4.2 - Find the image, rank, kernel, and nullity of the...Ch. 4.2 - Find the kernel and nullity of the transformation...Ch. 4.2 - Find the image and kernel of the transformation T...Ch. 4.2 - For the transformation T in Exercise 23, find the...Ch. 4.2 - For the transformation T in Exercise 42, find the...Ch. 4.2 - Find the image and kernel of the transformation T...Ch. 4.2 - Find the image and kernel of the transformation T...Ch. 4.2 - Define an isomorphism from P3 to 3 , if you can.Ch. 4.2 - Define an isomorphism from P3 to 22 , if you can.Ch. 4.2 - We will define a transformation T from nm to...Ch. 4.2 - Find the kernel and nullity of the linear...Ch. 4.2 - For which constants k is the linear transformation...Ch. 4.2 - For which constants k is the linear transformation...Ch. 4.2 - If matrix A is similar to B, is T(M)=AMMB an...Ch. 4.2 - For which real numbers co, c0,c1,...,cn is the...Ch. 4.2 - Prob. 71ECh. 4.2 - Prob. 72ECh. 4.2 - Prob. 73ECh. 4.2 - In Exercises 72 through 74, let Znbe the set of...Ch. 4.2 - Prob. 75ECh. 4.2 - Prob. 76ECh. 4.2 - Prob. 77ECh. 4.2 - Let + be the set of positive real numbers. On + we...Ch. 4.2 - Prob. 79ECh. 4.2 - Prob. 80ECh. 4.2 - Prob. 81ECh. 4.2 - Prob. 82ECh. 4.2 - Consider linear transformations T from V to W and...Ch. 4.2 - Prob. 84ECh. 4.3 - GOAL Use the concept of coordinates. Find the...Ch. 4.3 - GOAL Use the concept of coordinates. Find the...Ch. 4.3 - Do the polynomials...Ch. 4.3 - Consider the polynomials f(t)=t+1 and...Ch. 4.3 - Prob. 5ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Prob. 13ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 15ECh. 4.3 - Prob. 16ECh. 4.3 - Prob. 17ECh. 4.3 - Prob. 18ECh. 4.3 - Prob. 19ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 21ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 25ECh. 4.3 - Prob. 26ECh. 4.3 - Prob. 27ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 29ECh. 4.3 - Prob. 30ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 32ECh. 4.3 - In Exercises 5 through 40, find the matrix of the...Ch. 4.3 - Prob. 34ECh. 4.3 - Prob. 35ECh. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.3 - Prob. 38ECh. 4.3 - Prob. 39ECh. 4.3 - Prob. 40ECh. 4.3 - Prob. 41ECh. 4.3 - Prob. 42ECh. 4.3 - Prob. 43ECh. 4.3 - a. Find the change of basis matrix S from the...Ch. 4.3 - Prob. 45ECh. 4.3 - a. Find the change of basis matrix S from the...Ch. 4.3 - a. Find the change of basis matrix S from the...Ch. 4.3 - Prob. 48ECh. 4.3 - Prob. 49ECh. 4.3 - In Exercises 48 through 53, let V be the space...Ch. 4.3 - Prob. 51ECh. 4.3 - Prob. 52ECh. 4.3 - Prob. 53ECh. 4.3 - In Exercises 54 through 58, let V be the plane...Ch. 4.3 - Prob. 55ECh. 4.3 - Prob. 56ECh. 4.3 - Prob. 57ECh. 4.3 - Prob. 58ECh. 4.3 - Consider a linear transformation T from V to V...Ch. 4.3 - In the plane V defined by the equation 2x1+x22x3=0...Ch. 4.3 - Prob. 61ECh. 4.3 - Prob. 62ECh. 4.3 - Prob. 63ECh. 4.3 - Let V be the space of all upper triangular 22...Ch. 4.3 - Let V be the subspace of 22 spanned by the...Ch. 4.3 - Prob. 66ECh. 4.3 - Let V be the linear space of all functions of the...Ch. 4.3 - Consider the linear space V of all infinite...Ch. 4.3 - Consider a basis f1,...,fn , of Pn1.Let a1,...,an...Ch. 4.3 - Prob. 70ECh. 4.3 - Prob. 71ECh. 4.3 - In all parts of this problem, let V be the set of...Ch. 4.3 - Prob. 73ECh. 4 - The polynomials of degree less than 7 form a seven...Ch. 4 - Prob. 2ECh. 4 - Prob. 3ECh. 4 - Prob. 4ECh. 4 - The space 23 is five-dimensional.Ch. 4 - Prob. 6ECh. 4 - Prob. 7ECh. 4 - Prob. 8ECh. 4 - If W1 and W2 are subspaces of a linear space V,...Ch. 4 - If T is a linear transformation from P6 to 22 ,...Ch. 4 - Prob. 11ECh. 4 - Prob. 12ECh. 4 - Prob. 13ECh. 4 - All linear transformations from P3 to 22 are...Ch. 4 - If T is a linear transformation from V to V, then...Ch. 4 - Prob. 16ECh. 4 - Every polynomial of degree 3 can be expressed as a...Ch. 4 - a linear space V can be spanned by 10 elements,...Ch. 4 - Prob. 19ECh. 4 - There exists a 22 matrix A such that the space V...Ch. 4 - Prob. 21ECh. 4 - Prob. 22ECh. 4 - Prob. 23ECh. 4 - Prob. 24ECh. 4 - Prob. 25ECh. 4 - Prob. 26ECh. 4 - Prob. 27ECh. 4 - Prob. 28ECh. 4 - Prob. 29ECh. 4 - Prob. 30ECh. 4 - If W is a subspace of V, and if W is finite...Ch. 4 - Prob. 32ECh. 4 - Prob. 33ECh. 4 - Prob. 34ECh. 4 - Prob. 35ECh. 4 - Prob. 36ECh. 4 - Prob. 37ECh. 4 - Prob. 38ECh. 4 - Prob. 39ECh. 4 - Prob. 40ECh. 4 - Prob. 41ECh. 4 - The transformation D(f)=f from C to C is an...Ch. 4 - If T is a linear transformation from P4 to W with...Ch. 4 - The kernel of the linear transformation...Ch. 4 - If T is a linear transformation from V to V, then...Ch. 4 - If T is a linear transformation from P6 to P6 that...Ch. 4 - There exist invertible 22 matrices P and Q such...Ch. 4 - There exists a linear transformation from P6 to ...Ch. 4 - If f1,f2,f3 is a basis of a linear space V, and if...Ch. 4 - There exists a two-dimensional subspace of 22...Ch. 4 - The space P11 is isomorphic to 34 .Ch. 4 - If T is a linear transformation from V to W, and...Ch. 4 - If T is a linear transformation from V to 22 with...Ch. 4 - The function T(f(t))=ddt23t+4f(x)dx from P5 to P5...Ch. 4 - Any four-dimensional linear space has infinitely...Ch. 4 - If the matrix of a linear transformation T (with...Ch. 4 - If the image of a linear transformation T is...Ch. 4 - There exists a 22 matrix A such that the space of...Ch. 4 - If A, B, C, and D are noninvertible 22 matrices,...Ch. 4 - There exist two distinct three-dimensional...Ch. 4 - the elements f1,...,fn , (where f10 ) are linearly...Ch. 4 - There exists a 33 matrix P such that the linear...Ch. 4 - If f1,f2,f3,f4,f5 are elements of a linear space...Ch. 4 - There exists a linear transformation T from P6 to...Ch. 4 - If T is a linear transformation from V to W, and...Ch. 4 - If the matrix of a linear transformation T (with...Ch. 4 - Every three-dimensional subspace of 22 contains at...
Knowledge Booster
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
- (c) Let A = -1 3 -4 12 3 3 -9 (i) Find bases for row(A), col(A) and N(A). (ii) Determine the rank and nullity of A, and verify that the Rank-Nullity Theorem holds for the above matrix A.arrow_forward-(0)-(0)-(0) X1 = x2 = x3 = 1 (a) Show that the vectors X1, X2, X3 form a basis for R³. y= (b) Find the coordinate vector [y] B of y in the basis B = (x1, x2, x3).arrow_forwardLet A 1 - 13 (1³ ³) 3). (i) Compute A2, A3, A4. (ii) Show that A is invertible and find A-¹.arrow_forward
- Let H = {(a a12 a21 a22, | a1 + a2 = 0} . € R²x²: a11 + a22 (i) Show that H is a subspace of R2×2 (ii) Find a basis of H and determine dim H.arrow_forward2 5 A=1 2 -2 b=2 3 1 -1 3 (a) Calculate det(A). (b) Using (a), deduce that the system Ax = b where x = (x1, x2, x3) is consistent and determine x2 using Cramer's rule.arrow_forwardConsider the least squares problem Ax = b, where 12 -09-0 A 1 3 1 4 and b = (a) Write down the corresponding normal equations. (b) Determine the set of least squares solutions to the problem.arrow_forward
- The function f(x) is represented by the equation, f(x) = x³ + 8x² + x − 42. Part A: Does f(x) have zeros located at -7, 2, -3? Explain without using technology and show all work. Part B: Describe the end behavior of f(x) without using technology.arrow_forwardHow does the graph of f(x) = (x − 9)4 – 3 compare to the parent function g(x) = x²?arrow_forwardFind the x-intercepts and the y-intercept of the graph of f(x) = (x − 5)(x − 2)(x − 1) without using technology. Show all work.arrow_forward
- In a volatile housing market, the overall value of a home can be modeled by V(x) = 415x² - 4600x + 200000, where V represents the value of the home and x represents each year after 2020. Part A: Find the vertex of V(x). Show all work. Part B: Interpret what the vertex means in terms of the value of the home.arrow_forwardShow all work to solve 3x² + 5x - 2 = 0.arrow_forwardTwo functions are given below: f(x) and h(x). State the axis of symmetry for each function and explain how to find it. f(x) h(x) 21 5 4+ 3 f(x) = −2(x − 4)² +2 + -5 -4-3-2-1 1 2 3 4 5 -1 -2 -3 5arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Big Ideas Math A Bridge To Success Algebra 1: Stu...AlgebraISBN:9781680331141Author:HOUGHTON MIFFLIN HARCOURTPublisher:Houghton Mifflin Harcourt
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781337278461
Author:Ron Larson
Publisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Big Ideas Math A Bridge To Success Algebra 1: Stu...
Algebra
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:Houghton Mifflin Harcourt
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