Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN: 9781305658004
Author: Ron Larson
Publisher: Cengage Learning
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Chapter 6.2, Problem 12E

Finding the Kernel and Range In Exercises 11-18, define the linear transformation T by T ( x ) = A x . Find (a) the kernel of T and (b) the range of T .

A = [ 1 2 − 3 − 6 ]

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Solve the linear system of equations attached using Gaussian elimination (not Gauss-Jordan) and back subsitution. Remember that: A matrix is in row echelon form if Any row that consists only of zeros is at the bottom of the matrix. The first non-zero entry in each other row is 1. This entry is called aleading 1. The leading 1 of each row, after the first row, lies to the right of the leading 1 of the previous row.

Chapter 6 Solutions

Elementary Linear Algebra (MindTap Course List)

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For a fixed...Ch. 6.1 - Prob. 81ECh. 6.1 - Prob. 82ECh. 6.1 - Prob. 83ECh. 6.1 - Prob. 84ECh. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel of a Linear Transformation In...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel and Range In Exercises 11-18,...Ch. 6.2 - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.2 - Finding the Kernel, Nullity, Range, and RankIn...Ch. 6.2 - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.2 - Prob. 32ECh. 6.2 - Finding the Nullity and Describing the Kernel and...Ch. 6.2 - Prob. 34ECh. 6.2 - Prob. 35ECh. 6.2 - Finding the Nullity and Describing the Kernel and...Ch. 6.2 - Prob. 37ECh. 6.2 - Prob. 38ECh. 6.2 - Finding the Nullity and Describing the Kernel and...Ch. 6.2 - Prob. 40ECh. 6.2 - Finding the Nullity of a Linear Transformation In...Ch. 6.2 - Prob. 42ECh. 6.2 - Finding the Nullity of a Linear TransformationIn...Ch. 6.2 - Finding the Nullity of a Linear TransformationIn...Ch. 6.2 - Finding the Nullity of a Linear TransformationIn...Ch. 6.2 - Prob. 46ECh. 6.2 - Verifying That T Is One-to-One and Onto In...Ch. 6.2 - Verifying That T Is One-to-One and Onto In...Ch. 6.2 - Verifying That T Is One-to-One and Onto In...Ch. 6.2 - Prob. 50ECh. 6.2 - Prob. 51ECh. 6.2 - Prob. 52ECh. 6.2 - Prob. 53ECh. 6.2 - Determining Whether T Is One-to-One, Onto, or...Ch. 6.2 - Identify the zero element and standard basis for...Ch. 6.2 - Which vector spaces are isomorphic to R6? a M2,3 b...Ch. 6.2 - Calculus Define T:P4P3 by T(p)=p. What is the...Ch. 6.2 - Calculus Define T:P2R by T(p)=01p(x)dx What is the...Ch. 6.2 - Let T:R3R3 be the linear transformation that...Ch. 6.2 - CAPSTONE Let T:R4R3 be the linear transformation...Ch. 6.2 - Prob. 61ECh. 6.2 - Prob. 62ECh. 6.2 - Prob. 63ECh. 6.2 - Prob. 64ECh. 6.2 - Prob. 65ECh. 6.2 - Prob. 66ECh. 6.2 - Guided Proof Let B be an invertible nn matrix....Ch. 6.2 - Prob. 68ECh. 6.2 - Prob. 69ECh. 6.2 - Prob. 70ECh. 6.3 - The Standard Matrix for a Linear TransformationIn...Ch. 6.3 - The Standard Matrix for a Linear TransformationIn...Ch. 6.3 - The Standard Matrix for a Linear TransformationIn...Ch. 6.3 - The Standard Matrix for a Linear TransformationIn...Ch. 6.3 - The Standard Matrix for a Linear TransformationIn...Ch. 6.3 - The Standard Matrix for a Linear Transformation In...Ch. 6.3 - Finding the Image of a Vector In Exercises 7-10,...Ch. 6.3 - Finding the Image of a Vector In Exercises 7-10,...Ch. 6.3 - Finding the Image of a Vector In Exercises 7-10,...Ch. 6.3 - Finding the Image of a Vector In Exercises 7-10,...Ch. 6.3 - Finding the Standard Matrix and the ImageIn...Ch. 6.3 - Finding the Standard Matrix and the Image In...Ch. 6.3 - Finding the Standard Matrix and the Image In...Ch. 6.3 - Prob. 14ECh. 6.3 - Finding the Standard Matrix and the Image In...Ch. 6.3 - Finding the Standard Matrix and the ImageIn...Ch. 6.3 - Prob. 17ECh. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Prob. 20ECh. 6.3 - Finding the Standard Matrix and the Image In...Ch. 6.3 - Finding the Standard Matrix and the Image In...Ch. 6.3 - Finding the Standard Matrix and the Image In...Ch. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Finding Standard Matrices for CompositionsIn...Ch. 6.3 - Prob. 28ECh. 6.3 - Finding Standard Matrices for Compositions In...Ch. 6.3 - Finding Standard Matrices for Compositions In...Ch. 6.3 - Finding the Inverse of a Linear TransformationIn...Ch. 6.3 - Finding the Inverse of a Linear TransformationIn...Ch. 6.3 - Finding the Inverse of a Linear TransformationIn...Ch. 6.3 - Prob. 34ECh. 6.3 - Finding the Inverse of a linear TransformationIn...Ch. 6.3 - Finding the Inverse of a Linear Transformation In...Ch. 6.3 - Finding the Image Two Ways In Exercises 37-42,...Ch. 6.3 - Finding the Image Two Ways In Exercises 37-42,...Ch. 6.3 - Finding the Image Two Ways In Exercises 37-42,...Ch. 6.3 - Prob. 40ECh. 6.3 - Prob. 41ECh. 6.3 - Finding the Image Two Ways In Exercises 37-42,...Ch. 6.3 - Let T:P2P3 be the linear transformation T(p)=xp....Ch. 6.3 - Let T:P2P4 be the linear transformation T(p)=x2p....Ch. 6.3 - Calculus Let B={1,x,ex,xex} be a basis for a...Ch. 6.3 - Calculus Repeat Exercise 45 for...Ch. 6.3 - Calculus Use the matrix from Exercise 45 to...Ch. 6.3 - Prob. 48ECh. 6.3 - Calculus Let B={1,x,x2,x3} be a basis for P3, and...Ch. 6.3 - Prob. 50ECh. 6.3 - Define T:M2,3M3,2 by T(A)=AT. aFind the matrix for...Ch. 6.3 - Let T be a linear transformation T such that...Ch. 6.3 - True or False? In Exercises 53 and 54, determine...Ch. 6.3 - Prob. 54ECh. 6.3 - Prob. 55ECh. 6.3 - Prob. 56ECh. 6.3 - Prob. 57ECh. 6.3 - Writing Look back at theorem 4.19 and rephrase it...Ch. 6.4 - Finding a Matrix for a Linear Transformation In...Ch. 6.4 - Finding a Matrix for a Linear Transformation In...Ch. 6.4 - Prob. 3ECh. 6.4 - Finding a Matrix for a Linear Transformation In...Ch. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Finding a Matrix for a Linear Transformation In...Ch. 6.4 - Prob. 9ECh. 6.4 - Finding a Matrix for a Linear Transformation In...Ch. 6.4 - Prob. 11ECh. 6.4 - Prob. 12ECh. 6.4 - Prob. 13ECh. 6.4 - Repeat Exercise 13 for B={(1,1),(2,3)},...Ch. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Prob. 17ECh. 6.4 - Repeat Exercise 17 for...Ch. 6.4 - Similar Matrices In Exercises 19-22, use the...Ch. 6.4 - Similar Matrices In Exercises 19-22, use the...Ch. 6.4 - Similar Matrices In Exercises 19-22, use the...Ch. 6.4 - Similar Matrices In Exercises 19-22, use the...Ch. 6.4 - Diagonal Matrix for a Linear Transformation In...Ch. 6.4 - Diagonal Matrix for a Linear Transformation In...Ch. 6.4 - Proof Prove that if A and B are similar matrices,...Ch. 6.4 - Illustrate the result of exercise 25 using the...Ch. 6.4 - Prob. 27ECh. 6.4 - Prob. 28ECh. 6.4 - Prob. 29ECh. 6.4 - Prob. 30ECh. 6.4 - Prob. 31ECh. 6.4 - Prob. 32ECh. 6.4 - Prob. 33ECh. 6.4 - Prob. 34ECh. 6.4 - Prob. 35ECh. 6.4 - Proof Prove that if A and B are similar matrices...Ch. 6.4 - Prob. 37ECh. 6.4 - Prob. 38ECh. 6.4 - Prob. 39ECh. 6.4 - Prob. 40ECh. 6.4 - Prob. 41ECh. 6.4 - Prob. 42ECh. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6.5 - Prob. 10ECh. 6.5 - Prob. 11ECh. 6.5 - Prob. 12ECh. 6.5 - Prob. 13ECh. 6.5 - Prob. 14ECh. 6.5 - Prob. 15ECh. 6.5 - Prob. 16ECh. 6.5 - Prob. 17ECh. 6.5 - Prob. 18ECh. 6.5 - Prob. 19ECh. 6.5 - Prob. 20ECh. 6.5 - Finding Fixed Points of a Linear Transformation In...Ch. 6.5 - Finding Fixed Points of a Linear Transformation In...Ch. 6.5 - Prob. 23ECh. 6.5 - Prob. 24ECh. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6.5 - Prob. 27ECh. 6.5 - Prob. 28ECh. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Prob. 31ECh. 6.5 - Prob. 32ECh. 6.5 - Prob. 33ECh. 6.5 - Prob. 34ECh. 6.5 - Prob. 35ECh. 6.5 - Prob. 36ECh. 6.5 - Sketching an Image of a Rectangle In Exercises...Ch. 6.5 - Sketching an Image of a Rectangle In Exercises...Ch. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - Prob. 41ECh. 6.5 - Prob. 42ECh. 6.5 - Prob. 43ECh. 6.5 - Prob. 44ECh. 6.5 - Giving a Geometric Description In Exercises 45-50,...Ch. 6.5 - Prob. 46ECh. 6.5 - Prob. 47ECh. 6.5 - Prob. 48ECh. 6.5 - Prob. 49ECh. 6.5 - Giving a Geometric Description In Exercises 45-50,...Ch. 6.5 - Prob. 51ECh. 6.5 - Prob. 52ECh. 6.5 - Prob. 53ECh. 6.5 - Prob. 54ECh. 6.5 - Prob. 55ECh. 6.5 - Prob. 56ECh. 6.5 - Prob. 57ECh. 6.5 - Prob. 58ECh. 6.5 - Prob. 59ECh. 6.5 - Prob. 60ECh. 6.5 - Prob. 61ECh. 6.5 - Prob. 62ECh. 6.5 - Prob. 63ECh. 6.5 - Prob. 64ECh. 6.5 - Prob. 65ECh. 6.5 - Prob. 66ECh. 6.5 - Prob. 67ECh. 6.5 - Prob. 68ECh. 6.5 - Prob. 69ECh. 6.5 - Determining a matrix to produce a pair of rotation...Ch. 6.5 - Prob. 71ECh. 6.5 - Prob. 72ECh. 6.CR - Prob. 1CRCh. 6.CR - Finding an Image and a PreimageIn Exercises 1-6,...Ch. 6.CR - Finding an Image and a PreimageIn Exercises 1-6,...Ch. 6.CR - Prob. 4CRCh. 6.CR - Finding an Image and a PreimageIn Exercises 1-6,...Ch. 6.CR - Prob. 6CRCh. 6.CR - Linear Transformations and Standard Matrices In...Ch. 6.CR - Prob. 8CRCh. 6.CR - Linear Transformations and Standard MatricesIn...Ch. 6.CR - Linear Transformations and Standard MatricesIn...Ch. 6.CR - Linear Transformations and Standard MatricesIn...Ch. 6.CR - Prob. 12CRCh. 6.CR - Linear Transformations and Standard MatricesIn...Ch. 6.CR - Linear Transformations and Standard MatricesIn...Ch. 6.CR - Linear Transformations and Standard MatricesIn...Ch. 6.CR - Prob. 16CRCh. 6.CR - Linear Transformations and Standard MatricesIn...Ch. 6.CR - Prob. 18CRCh. 6.CR - Let T be a linear transformation from R2 into R2...Ch. 6.CR - Let T be a linear transformation from R3 into R...Ch. 6.CR - Let T be a linear transformation from R2 into R2...Ch. 6.CR - Let T be a linear transformation from R2 into R2...Ch. 6.CR - Linear Transformation Given by a Matrix In...Ch. 6.CR - Linear Transformation Given by a Matrix In...Ch. 6.CR - Linear Transformation Given by a Matrix In...Ch. 6.CR - Linear Transformation Given by a Matrix In...Ch. 6.CR - Linear Transformation Given by a Matrix In...Ch. 6.CR - Linear Transformation Given by a MatrixIn...Ch. 6.CR - Use the standard matrix for counterclockwise...Ch. 6.CR - Rotate the triangle in Exercise 29...Ch. 6.CR - Finding the Kernel and Range In Exercises 31-34,...Ch. 6.CR - Finding the Kernel and Range In Exercises 31-34,...Ch. 6.CR - Finding the Kernel and Range In Exercises 31-34,...Ch. 6.CR - Finding the Kernel and Range In Exercises 31-34,...Ch. 6.CR - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.CR - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.CR - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.CR - Finding the Kernel, Nullity, Range, and Rank In...Ch. 6.CR - For T:R5R3 and nullity(T)=2, find rank(T).Ch. 6.CR - For T:P5P3 and nullity(T)=4, find rank(T).Ch. 6.CR - For T:P4R5, and rank (T)=3, find nullity (T).Ch. 6.CR - Prob. 42CRCh. 6.CR - Prob. 43CRCh. 6.CR - Prob. 44CRCh. 6.CR - Prob. 45CRCh. 6.CR - Prob. 46CRCh. 6.CR - Finding Standard Matrices for Compositions In...Ch. 6.CR - Prob. 48CRCh. 6.CR - Prob. 49CRCh. 6.CR - Prob. 50CRCh. 6.CR - Finding the Inverse of a Linear Transformation In...Ch. 6.CR - Finding the Inverse of a Linear Transformation In...Ch. 6.CR - One-to-One, Onto, and Invertible Transformations...Ch. 6.CR - One-to-One, Onto, and Invertible Transformations...Ch. 6.CR - One-to-One, Onto, and Invertible Transformations...Ch. 6.CR - One-to-One, Onto, and Invertible Transformations...Ch. 6.CR - Finding the Image Two Ways InExercises 57 and 58,...Ch. 6.CR - Finding the Image Two Ways In Exercises 57 and 58,...Ch. 6.CR - Finding a Matrix for a Linear Transformation In...Ch. 6.CR - Prob. 60CRCh. 6.CR - Prob. 61CRCh. 6.CR - Prob. 62CRCh. 6.CR - Prob. 63CRCh. 6.CR - Prob. 64CRCh. 6.CR - Prob. 65CRCh. 6.CR - Prob. 66CRCh. 6.CR - Sum of Two Linear Transformations In Exercises 67...Ch. 6.CR - Prob. 68CRCh. 6.CR - Prob. 69CRCh. 6.CR - Prob. 70CRCh. 6.CR - Let V be an inner product space. For a fixed...Ch. 6.CR - Calculus Let B={1,x,sinx,cosx} be a basis for a...Ch. 6.CR - Prob. 73CRCh. 6.CR - Prob. 74CRCh. 6.CR - Prob. 75CRCh. 6.CR - Prob. 76CRCh. 6.CR - Prob. 77CRCh. 6.CR - Prob. 78CRCh. 6.CR - Prob. 79CRCh. 6.CR - Prob. 80CRCh. 6.CR - Prob. 81CRCh. 6.CR - Prob. 82CRCh. 6.CR - Prob. 83CRCh. 6.CR - Prob. 84CRCh. 6.CR - Prob. 85CRCh. 6.CR - Prob. 86CRCh. 6.CR - Prob. 87CRCh. 6.CR - Prob. 88CRCh. 6.CR - Prob. 89CRCh. 6.CR - Prob. 90CRCh. 6.CR - Prob. 91CRCh. 6.CR - Prob. 92CRCh. 6.CR - Prob. 93CRCh. 6.CR - Prob. 94CRCh. 6.CR - Prob. 95CRCh. 6.CR - Prob. 96CRCh. 6.CR - Prob. 97CRCh. 6.CR - Prob. 98CRCh. 6.CR - True or False? In Exercises 99-102, determine...Ch. 6.CR - True or False? In Exercises 99-102, determine...Ch. 6.CR - Prob. 101CRCh. 6.CR - Prob. 102CR
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