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CONTENTS

Chapter 1	Systems of Linear Equations	1
1.1	Solving Linear Systems	2
1.2	Geometric Perspectives on Linear Systems	18
1.3A	Solving Linear Systems Using Maple	26
1.3B	Solving Linear Systems Using Mathematica	36
1.4	Curve Fitting and Temperature Distribution -- Application	45
Chapter 2	Vectors	56
2.1	Geometry of Vectors	57
2.2	Linear Combinations of Vectors	71
2.3	Decomposing the Solution of a Linear System	84
2.4	Linear Independence of Vectors	92
2.5	Theory of Vector Concepts	106
Chapter 3	Matrix Algebra	113
3.1	Product of a Matrix and a Vector	114
3.2	Matrix Multiplication	125
3.3	Rules of Matrix Algebra	140
3.4	Markov Chains -- Application	147
3.5	Inverse of a Matrix	161
3.6	Theory of Matrix Inverses	175
3.7	Cryptology -- Application	180
Chapter 4	Linear Transformations	193
4.1	Introduction to Matrix Transformations	194
4.2	Geometry of Matrix Transformations of the Plane	198
4.3	Geometry of Matrix Transformations of 3-Space	220
4.4	Linear Transformations	232
4.5	Computer Graphics -- Application	237
Chapter 5	Vector Spaces	255
5.1	Subspaces of $\mathbb{R}^{n}$	256
5.2	Basis and Dimension	262
5.3	Theory of Basis and Dimension	272
5.4	Subspaces Associated with a Matrix	275
5.5	Theory of Subspaces Associated with a Matrix	285
5.6	Loops and Spanning Trees -- Application	289
5.7	Abstract Vector Spaces	297
Chapter 6	Determinants	306
6.1	Determinants and Cofactors	307
6.2	Properties of Determinants	311
6.3	Theory of Determinants	322
Chapter 7	Eigenvalues and Eigenvectors	328
7.1	Introduction to Eigenvalues and Eigenvectors	329
7.2	The Characteristic Polynomial	343
7.3	Discrete Dynamical Systems -- Application	356
7.4	Diagonalization and Similar Matrices	373
7.5	Theory of Eigenvalues and Eigenvectors	389
7.6	Systems of Linear Differential Equations -- Application	395
7.7	Complex Numbers and Complex Vectors	407
7.8	Complex Eigenvalues and Eigenvectors	411
Chapter 8	Orthogonality	431
8.1	Dot Product and Orthogonal Vectors	432
8.2	Orthogonal Projections in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$	437
8.3	Orthogonal Projections and Orthogonal Bases in $\mathbb{R}^{n}$	447
8.4	Theory of Orthogonality	456
8.5	Least-Squares Solutions -- Application	463
8.6	Weighted Least-Squares and Inner Products on $\mathbb{R}^{n}$	475
8.7	Approximation of Functions and Integral Inner Products	483
8.8	Inner Product Spaces	496
Appendix A	Glossary of Linear Algebra Definitions	503
Appendix B	Linear Algebra Theorems	510
Appendix C	Advice for Using Maple with Visual Linear Algebra	519
Appendix D	Commands Used in Maple Tutorials	521
Appendix E	Advice for Using Mathematica with Visual Linear Algebra	527
Appendix F	Commands Used in Mathematica Tutorials	530
Appendix G	Answers and Hints for Selected Pencil and Paper Problems	537
Index		545

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