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Introductory Notes in Linear Algebra for the Engineers
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  • Course
  • B.SC. MATHEMATICS (MATH121)
  • Institution
  • KANNUR UNIVERSITY
  • Book
  • Linear Algebra as an Introduction to Abstract Mathematics

Lecture notes study book Linear Algebra as an Introduction to Abstract Mathematics of Isaiah Lankham, Bruno Nachtergaele, Anne Schilling - ISBN: 9789814723794 (ITS LECTURE NOTES)

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  • February 12, 2022
  • 251
  • 2021/2022
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  • introductory notes in linear algebra for the engineers

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  • KANNUR UNIVERSITY
  • B.SC. MATHEMATICS (MATH121)
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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

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