Solution Manual For
Introduction to Linear Algebra with Applications 1st Edition by Jim DeFranza
Chapter 1-6 With Preliminaries
Contents
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams, case
studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making skills, as
well as the ability to evaluate various business alternatives.Quantitative
1 Systems of Linear Equations and Matrices 1
Exercise Set 1.1 Systems of Linear Equations ......................................................................................................1
Exercise Set 1.2 Matrices and Elementary Row Operations ...............................................................................7
Exercise Set 1.3 Matrix Algebra ........................................................................................................................ 11
Exercise Set 1.4 The Inverse of a Matrix .......................................................................................................... 15
Exercise Set 1.5 Matrix Equations ....................................................................................................................... 19
Exercise Set 1.6 Determinants .............................................................................................................................. 22
Exercise Set 1.7 Elementary Matrices and LU Factorization ........................................................................ 27
Exercise Set 1.8 Applications of Systems of Linear Equations ..................................................................... 32
Review Exercises...................................................................................................................................................... 37
Chapter Test .......................................................................................................................................................... 40
2 Linear Combinations and Linear Independence 42
n
Exercise Set 2.1 Vectors in R .............................................................................................................................................................................. 42
Exercise Set 2.2 Linear Combinations.................................................................................................................. 46
Exercise Set 2.3 Linear Independence ................................................................................................................. 51
Review Exercises...................................................................................................................................................... 55
Chapter Test .......................................................................................................................................................... 58
3 Vector Spaces 60
Exercise Set 3.1 Definition of a Vector Space .................................................................................................... 60
Exercise Set 3.2 Subspaces ................................................................................................................................... 64
Exercise Set 3.3 Basis and Dimension ................................................................................................................. 71
Exercise Set 3.4 Coordinates and Change of Basis ............................................................................................ 77
Exercise Set 3.5 Application: Differential Equations ......................................................................................... 81
Review Exercises...................................................................................................................................................... 82
Chapter Test .......................................................................................................................................................... 86
4 Linear Transformations 88
Exercise Set 4.1 Linear Transformations ............................................................................................................. 88
Exercise Set 4.2 The Null Space and Range ..................................................................................................... 93
Exercise Set 4.3 Isomorphisms.............................................................................................................................. 98
Exercise Set 4.4 Matrix Transformation of a Linear Transformation............................................................. 101
Exercise Set 4.5 Similarity ................................................................................................................................... 106
Exercise Set 4.6 Application: Computer Graphics ........................................................................................ 110
Review Exercises.................................................................................................................................................... 113
Chapter Test ........................................................................................................................................................ 116
5 Eigenvalues and Eigenvectors 118
Exercise Set 5.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Exercise Set 5.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Exercise Set 5.3 Application: Systems of Linear Differential Equations . . . . . . . . . . . . . . . . . 128
Exercise Set 5.4 Application: Markov Chains ................................................................................................ 130
i
, ii CONTENTS
Review Exercises.................................................................................................................................................... 132
Chapter Test ........................................................................................................................................................ 135
6 Inner Product Spaces 137
n
Exercise Set 6.1 The Dot Product on R ................................................................................................................................................... 137
Exercise Set 6.2 Inner Product Spaces ............................................................................................................. 140
Exercise Set 6.3 Orthonormal Bases ................................................................................................................. 144
Exercise Set 6.4 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Exercise Set 6.5 Application: Least Squares Approximation . . . . . . . . . . . . . . . . . . . . . . . 157
Exercise Set 6.6 Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 161
Exercise Set 6.7 Application: Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Exercise Set 6.8 Application: Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 166
Review Exercises .................................................................................................................................................. 168
Chapter Test ........................................................................................................................................................ 171
A Preliminaries 173
Exercise Set A.1 Algebra of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Exercise Set A.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Exercise Set A.3 Techniques of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Exercise Set A.4 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business
exams, case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make
strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making skills, as wel
as the ability to evaluate various business alternatives.Quantitative
, 1.1 Systems of Linear Equations 1
Solutions to All Exercises
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams, case
studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business ExamsCritical
Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making skills, as well as the ability
to evaluate various business alternatives.Quantitative
1 Systems of Linear Equations and
Matrices
Exercise Set 1.1
In Section 1.1 of the text, Gaussian Elimination is used to solve a linear system. This procedure utilizes
three operations that when applied to a linear system result in a new system that is equivalent to the original.
Equivalent means that the linear systems have the same solutions. The three operations are:
• Interchange two equations.
• Multiply any equation by a nonzero constant.
• Add a multiple of one equation to another.
When used judiciously these three operations allow us to reduce a linear system to a triangular linear system,
which can be solved. A linear system is consistent if there is at least one solution and is inconsistent if there
are no solutions. Every linear system has either a unique solution, infinitely many solutions or no solutions.
For example, the triangular linear systems
x1 − x2 + x3 =2 x1 − 2x2 + x3 =2 ⎪2x1 + x3 =1
x2 − 2x3 = −1 , −x2 + 2x3 = −3 ,
⎪
⎩ ⎪
⎩ ⎪
⎩
x3 = 2 0 =4
have a unique solution, infinitely many solutions, and no solutions, respectively. In the second linear system,
the variable x3 is a free variable, and once assigned any real number the values of x1 and x2 are determined.
In this way the linear system has infinitely many solutions. If a linear system has the same form as the second
system, but also has the additional equation 0 = 0, then the linear system will still have free variables. The
third system is inconsistent since the last equation 0 = 4 is impossible. In some cases, the conditions on the
right hand side of a linear system are not specified. Consider for example, the linear system
−x1 − x2 =a −x1 − x2 =a
2x1 + 2x2 + x3 = b which is equivalent to x3 = b + 2a .
⎪
⎩ −−−−−−−−−−−−−−−−→ ⎪ ⎩
2x3 = c 0 = c − 2b − 4a
This linear system is consistent only for values a, b and c such that c − 2b − 4a = 0.
Solutions to Exercises
1. Applying the given operations we obtain the equivalent triangular system
x1 − x2 − 2x3 =3 x1 − x2 − 2x3 =3
−x1 + 2x2 + 3x3 = 1 E1 + E2 → E2 x2 + x3 =4 (−2)E1 + E3 → E3
⎪
⎩ −−−−−−−−−−→ ⎪
⎩ −−−−−−−−−−−−−−→
2x 1 − 2x2 − 2x3 = −2 2x 1 − 2x2 − 2x3 = −2
, 2 Chapter 1 Systems of Linear Equations and Matrices
x2 + x3 =4 . Using back substitution, the linear system has the unique solution
⎪
⎩
2x3 = −8
x1 = 3, x2 = 8, x3 = −4.
2. Applying the given operations we obtain the equivalent triangular system
2x1 − 2x2 − x3 = −3 x1 − 3x2 + x3 = −2
x1 − 3x2 + x3 = −2 E1 ↔ E2 2x1 − 2x2 − x3 = −3 (−2)E1 + E2 → E2
⎪
⎩ −−−−−−→ ⎪
⎩ −−−−−−−−−−−−−−→
x 1 − 2x2 =2 x 1 − 2x2 =2
x1 − 3x2 + x3 = −2 x1 − 3x2 + x3 = −2
4x2 − 3x3 = 1 (−1)E1 + E3 → E3 4x2 − 3x3 = 1 E2 ↔ E3
⎪ −−−−−−−−−−−−−−→ ⎪⎩ −−−−−−→
⎩ x2 − x3 =4
x 1 − 2x2 =2
x1 − 3x2 + x3 = −2 x1 − 3x2 + x3 = −2
x2 − x3 = 4 (−4)E2 + E3 → E3 x2 − x3 =4 .
⎪
⎩ −−−−−−−−−−−−−−→ ⎪⎩
4x2 − 3x3 = 1 x3 = −15
Using back substitution, the linear system has the unique solution x1 = 20, x2 = 11, x3 = 15.
3. Applying the given operations we obtain the equivalent triangular system
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make
strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in
Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-
making skills, as well as the ability to evaluate various business alternatives.Quantitative
x1 + 3x4 =2 x1 + 3x4 =2
⎪
⎨
x1 + x2 + 4x4 =3 x2 + x4 =1
1 + E2 → E2
2x1 + x3 + 8x4 = 3 −−−−−−−−−−−−−−→ 2x1 + x3 + 8x4 = 3
⎩
x1 + x2 + x3 + 6x4 = 2 x1 + x2 + x3 + 6x4 = 2
x1 + 3x4 =2
⎪
⎨ x2 + x4 =1
(−2)E1 + E3 → E3 (−1)E1 + E4 → E4
−−−−−−−−−−−−−−→ +x3 + 2x4 = −1 −−−−−−−−−−−−−−→
x1 + x2 + x3 + 6x4 = 2
x1 + 3x4 =2 x1 + 3x4 = 2
⎪
⎪
⎪ +x3 + 2x4 = −1 −−−−−−−−−−−−−−→ ⎪
⎪ x3 + 2x4 = −1
⎩ ⎩
x2 + x3 + 3x4 = 0 x3 + 2x4 = −1
x1 + 3x4 = 2
⎪
⎨
(−1)E3 + E4 → E4 x2 + x4 =1
.
−−−−−−−−−−−−−−→ ⎪ x3 + 2x4 = −1
0 =0
The final triangular linear system has more variables than equations, that is, there is a free variable. As a
result there are infinitely many solutions. Specifically, using back substitution, the solutions are given by
x1 = 2 3x4, x2 = 1 x4, x3 = 1 2x4, x4 R.
4. Applying the given operations we obtain the equivalent triangular system
x1 + x3 = −2 x1 + x3 = −2
x1 + x2 + 4x3 = −1 (−1)E1 + E2 → E2 x2 + 3x3 = 1 (−2)E1 + E3 → E3
⎪
⎩ −−−−−−−−−−−−−−→ ⎪⎩ −−−−−−−−−−−−−−→
2 x 1 + 2x3 + x4 = −1 2x1 + 2x3 + x4 = −1
Introduction to Linear Algebra with Applications 1st Edition by Jim DeFranza
Chapter 1-6 With Preliminaries
Contents
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams, case
studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making skills, as
well as the ability to evaluate various business alternatives.Quantitative
1 Systems of Linear Equations and Matrices 1
Exercise Set 1.1 Systems of Linear Equations ......................................................................................................1
Exercise Set 1.2 Matrices and Elementary Row Operations ...............................................................................7
Exercise Set 1.3 Matrix Algebra ........................................................................................................................ 11
Exercise Set 1.4 The Inverse of a Matrix .......................................................................................................... 15
Exercise Set 1.5 Matrix Equations ....................................................................................................................... 19
Exercise Set 1.6 Determinants .............................................................................................................................. 22
Exercise Set 1.7 Elementary Matrices and LU Factorization ........................................................................ 27
Exercise Set 1.8 Applications of Systems of Linear Equations ..................................................................... 32
Review Exercises...................................................................................................................................................... 37
Chapter Test .......................................................................................................................................................... 40
2 Linear Combinations and Linear Independence 42
n
Exercise Set 2.1 Vectors in R .............................................................................................................................................................................. 42
Exercise Set 2.2 Linear Combinations.................................................................................................................. 46
Exercise Set 2.3 Linear Independence ................................................................................................................. 51
Review Exercises...................................................................................................................................................... 55
Chapter Test .......................................................................................................................................................... 58
3 Vector Spaces 60
Exercise Set 3.1 Definition of a Vector Space .................................................................................................... 60
Exercise Set 3.2 Subspaces ................................................................................................................................... 64
Exercise Set 3.3 Basis and Dimension ................................................................................................................. 71
Exercise Set 3.4 Coordinates and Change of Basis ............................................................................................ 77
Exercise Set 3.5 Application: Differential Equations ......................................................................................... 81
Review Exercises...................................................................................................................................................... 82
Chapter Test .......................................................................................................................................................... 86
4 Linear Transformations 88
Exercise Set 4.1 Linear Transformations ............................................................................................................. 88
Exercise Set 4.2 The Null Space and Range ..................................................................................................... 93
Exercise Set 4.3 Isomorphisms.............................................................................................................................. 98
Exercise Set 4.4 Matrix Transformation of a Linear Transformation............................................................. 101
Exercise Set 4.5 Similarity ................................................................................................................................... 106
Exercise Set 4.6 Application: Computer Graphics ........................................................................................ 110
Review Exercises.................................................................................................................................................... 113
Chapter Test ........................................................................................................................................................ 116
5 Eigenvalues and Eigenvectors 118
Exercise Set 5.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Exercise Set 5.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Exercise Set 5.3 Application: Systems of Linear Differential Equations . . . . . . . . . . . . . . . . . 128
Exercise Set 5.4 Application: Markov Chains ................................................................................................ 130
i
, ii CONTENTS
Review Exercises.................................................................................................................................................... 132
Chapter Test ........................................................................................................................................................ 135
6 Inner Product Spaces 137
n
Exercise Set 6.1 The Dot Product on R ................................................................................................................................................... 137
Exercise Set 6.2 Inner Product Spaces ............................................................................................................. 140
Exercise Set 6.3 Orthonormal Bases ................................................................................................................. 144
Exercise Set 6.4 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Exercise Set 6.5 Application: Least Squares Approximation . . . . . . . . . . . . . . . . . . . . . . . 157
Exercise Set 6.6 Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 161
Exercise Set 6.7 Application: Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Exercise Set 6.8 Application: Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 166
Review Exercises .................................................................................................................................................. 168
Chapter Test ........................................................................................................................................................ 171
A Preliminaries 173
Exercise Set A.1 Algebra of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Exercise Set A.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Exercise Set A.3 Techniques of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Exercise Set A.4 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business
exams, case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make
strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making skills, as wel
as the ability to evaluate various business alternatives.Quantitative
, 1.1 Systems of Linear Equations 1
Solutions to All Exercises
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams, case
studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business ExamsCritical
Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making skills, as well as the ability
to evaluate various business alternatives.Quantitative
1 Systems of Linear Equations and
Matrices
Exercise Set 1.1
In Section 1.1 of the text, Gaussian Elimination is used to solve a linear system. This procedure utilizes
three operations that when applied to a linear system result in a new system that is equivalent to the original.
Equivalent means that the linear systems have the same solutions. The three operations are:
• Interchange two equations.
• Multiply any equation by a nonzero constant.
• Add a multiple of one equation to another.
When used judiciously these three operations allow us to reduce a linear system to a triangular linear system,
which can be solved. A linear system is consistent if there is at least one solution and is inconsistent if there
are no solutions. Every linear system has either a unique solution, infinitely many solutions or no solutions.
For example, the triangular linear systems
x1 − x2 + x3 =2 x1 − 2x2 + x3 =2 ⎪2x1 + x3 =1
x2 − 2x3 = −1 , −x2 + 2x3 = −3 ,
⎪
⎩ ⎪
⎩ ⎪
⎩
x3 = 2 0 =4
have a unique solution, infinitely many solutions, and no solutions, respectively. In the second linear system,
the variable x3 is a free variable, and once assigned any real number the values of x1 and x2 are determined.
In this way the linear system has infinitely many solutions. If a linear system has the same form as the second
system, but also has the additional equation 0 = 0, then the linear system will still have free variables. The
third system is inconsistent since the last equation 0 = 4 is impossible. In some cases, the conditions on the
right hand side of a linear system are not specified. Consider for example, the linear system
−x1 − x2 =a −x1 − x2 =a
2x1 + 2x2 + x3 = b which is equivalent to x3 = b + 2a .
⎪
⎩ −−−−−−−−−−−−−−−−→ ⎪ ⎩
2x3 = c 0 = c − 2b − 4a
This linear system is consistent only for values a, b and c such that c − 2b − 4a = 0.
Solutions to Exercises
1. Applying the given operations we obtain the equivalent triangular system
x1 − x2 − 2x3 =3 x1 − x2 − 2x3 =3
−x1 + 2x2 + 3x3 = 1 E1 + E2 → E2 x2 + x3 =4 (−2)E1 + E3 → E3
⎪
⎩ −−−−−−−−−−→ ⎪
⎩ −−−−−−−−−−−−−−→
2x 1 − 2x2 − 2x3 = −2 2x 1 − 2x2 − 2x3 = −2
, 2 Chapter 1 Systems of Linear Equations and Matrices
x2 + x3 =4 . Using back substitution, the linear system has the unique solution
⎪
⎩
2x3 = −8
x1 = 3, x2 = 8, x3 = −4.
2. Applying the given operations we obtain the equivalent triangular system
2x1 − 2x2 − x3 = −3 x1 − 3x2 + x3 = −2
x1 − 3x2 + x3 = −2 E1 ↔ E2 2x1 − 2x2 − x3 = −3 (−2)E1 + E2 → E2
⎪
⎩ −−−−−−→ ⎪
⎩ −−−−−−−−−−−−−−→
x 1 − 2x2 =2 x 1 − 2x2 =2
x1 − 3x2 + x3 = −2 x1 − 3x2 + x3 = −2
4x2 − 3x3 = 1 (−1)E1 + E3 → E3 4x2 − 3x3 = 1 E2 ↔ E3
⎪ −−−−−−−−−−−−−−→ ⎪⎩ −−−−−−→
⎩ x2 − x3 =4
x 1 − 2x2 =2
x1 − 3x2 + x3 = −2 x1 − 3x2 + x3 = −2
x2 − x3 = 4 (−4)E2 + E3 → E3 x2 − x3 =4 .
⎪
⎩ −−−−−−−−−−−−−−→ ⎪⎩
4x2 − 3x3 = 1 x3 = −15
Using back substitution, the linear system has the unique solution x1 = 20, x2 = 11, x3 = 15.
3. Applying the given operations we obtain the equivalent triangular system
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make
strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in
Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-
making skills, as well as the ability to evaluate various business alternatives.Quantitative
x1 + 3x4 =2 x1 + 3x4 =2
⎪
⎨
x1 + x2 + 4x4 =3 x2 + x4 =1
1 + E2 → E2
2x1 + x3 + 8x4 = 3 −−−−−−−−−−−−−−→ 2x1 + x3 + 8x4 = 3
⎩
x1 + x2 + x3 + 6x4 = 2 x1 + x2 + x3 + 6x4 = 2
x1 + 3x4 =2
⎪
⎨ x2 + x4 =1
(−2)E1 + E3 → E3 (−1)E1 + E4 → E4
−−−−−−−−−−−−−−→ +x3 + 2x4 = −1 −−−−−−−−−−−−−−→
x1 + x2 + x3 + 6x4 = 2
x1 + 3x4 =2 x1 + 3x4 = 2
⎪
⎪
⎪ +x3 + 2x4 = −1 −−−−−−−−−−−−−−→ ⎪
⎪ x3 + 2x4 = −1
⎩ ⎩
x2 + x3 + 3x4 = 0 x3 + 2x4 = −1
x1 + 3x4 = 2
⎪
⎨
(−1)E3 + E4 → E4 x2 + x4 =1
.
−−−−−−−−−−−−−−→ ⎪ x3 + 2x4 = −1
0 =0
The final triangular linear system has more variables than equations, that is, there is a free variable. As a
result there are infinitely many solutions. Specifically, using back substitution, the solutions are given by
x1 = 2 3x4, x2 = 1 x4, x3 = 1 2x4, x4 R.
4. Applying the given operations we obtain the equivalent triangular system
x1 + x3 = −2 x1 + x3 = −2
x1 + x2 + 4x3 = −1 (−1)E1 + E2 → E2 x2 + 3x3 = 1 (−2)E1 + E3 → E3
⎪
⎩ −−−−−−−−−−−−−−→ ⎪⎩ −−−−−−−−−−−−−−→
2 x 1 + 2x3 + x4 = −1 2x1 + 2x3 + x4 = −1
