Arkansas Tech University
Department of Mathematics
Introductory Notes in Linear Algebra
for the Engineers
Marcel B. Finan
c All Rights Reserved
,2
,Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Linear Systems of Equations 7
1. Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 8
2. Equivalent Systems and Elementary Row Operations: The Elim-
ination Method . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Solving Linear Systems Using Augmented Matrices . . . . . . . . 20
4. Echelon Form and Reduced Echelon Form: Gaussian Elimination 27
5. Echelon Forms and Solutions to Linear Systems . . . . . . . . . 39
6. Homogeneous Systems of Linear Equations . . . . . . . . . . . . 46
Matrices 53
7. Matrices and Matrix Operations . . . . . . . . . . . . . . . . . . 54
8. Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 64
9. The Inverse of a Square Matrix . . . . . . . . . . . . . . . . . . . 72
10. Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . 78
11. Finding A−1 Using Elementary Matrices . . . . . . . . . . . . . 85
Determinants 93
12. Determinants by Cofactor Expansion . . . . . . . . . . . . . . . 94
13. Evaluating Determinants by Row Reduction . . . . . . . . . . . 101
14. Properties of the Determinant . . . . . . . . . . . . . . . . . . . 108
15. Finding A−1 Using Cofactor Expansions . . . . . . . . . . . . . 111
16. Application of Determinants to Systems: Cramer’s Rule . . . . 117
The Theory of Vector Spaces 123
17. Vector Spaces and Subspaces . . . . . . . . . . . . . . . . . . . 124
18. Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . 130
3
, 4 CONTENTS
Eigenvalues and Eigenvectors 139
19. The Eigenvalues of a Square Matrix . . . . . . . . . . . . . . . 140
20. Finding Eigenvectors and Eigenspaces . . . . . . . . . . . . . . 152
21. Diagonalization of a Matrix . . . . . . . . . . . . . . . . . . . . 164
Linear Transformations 171
22. An Example of Motivation . . . . . . . . . . . . . . . . . . . . . 172
23. Linear Transformation: Definition and Elementary Properties . 175
24. Kernel and Range of a Linear Transformation . . . . . . . . . . 182
25. Matrix Representation of a Linear Transformation . . . . . . . 193
Answer Key 203
Department of Mathematics
Introductory Notes in Linear Algebra
for the Engineers
Marcel B. Finan
c All Rights Reserved
,2
,Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Linear Systems of Equations 7
1. Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 8
2. Equivalent Systems and Elementary Row Operations: The Elim-
ination Method . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Solving Linear Systems Using Augmented Matrices . . . . . . . . 20
4. Echelon Form and Reduced Echelon Form: Gaussian Elimination 27
5. Echelon Forms and Solutions to Linear Systems . . . . . . . . . 39
6. Homogeneous Systems of Linear Equations . . . . . . . . . . . . 46
Matrices 53
7. Matrices and Matrix Operations . . . . . . . . . . . . . . . . . . 54
8. Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 64
9. The Inverse of a Square Matrix . . . . . . . . . . . . . . . . . . . 72
10. Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . 78
11. Finding A−1 Using Elementary Matrices . . . . . . . . . . . . . 85
Determinants 93
12. Determinants by Cofactor Expansion . . . . . . . . . . . . . . . 94
13. Evaluating Determinants by Row Reduction . . . . . . . . . . . 101
14. Properties of the Determinant . . . . . . . . . . . . . . . . . . . 108
15. Finding A−1 Using Cofactor Expansions . . . . . . . . . . . . . 111
16. Application of Determinants to Systems: Cramer’s Rule . . . . 117
The Theory of Vector Spaces 123
17. Vector Spaces and Subspaces . . . . . . . . . . . . . . . . . . . 124
18. Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . 130
3
, 4 CONTENTS
Eigenvalues and Eigenvectors 139
19. The Eigenvalues of a Square Matrix . . . . . . . . . . . . . . . 140
20. Finding Eigenvectors and Eigenspaces . . . . . . . . . . . . . . 152
21. Diagonalization of a Matrix . . . . . . . . . . . . . . . . . . . . 164
Linear Transformations 171
22. An Example of Motivation . . . . . . . . . . . . . . . . . . . . . 172
23. Linear Transformation: Definition and Elementary Properties . 175
24. Kernel and Range of a Linear Transformation . . . . . . . . . . 182
25. Matrix Representation of a Linear Transformation . . . . . . . 193
Answer Key 203
