Linear Algebra: A Geometric Approach, Second Edition, presents the standard computational aspects of linear algebra and includes a variety of intriguing interesting applications that would be interesting to motivate science and engineering students, as well as help mathematics students make the transition to more abstract advanced courses. The text guides students on how to think about mathematical concepts and write rigorous mathematical arguments.
Chapter 1: Vectors and Matrices 1
1: Vectors 1
2: Dot Product 18
3: Hyperplanes in Rn 28
4: Systems of Linear Equations and Gaussian Elimination 36
5: The Theory of Linear Systems 53
6: Some Applications 64
Chapter 2: Matrix Algebra 81
1: Matrix Operations 81
2: Linear Transformations: An Introduction 91
3: Inverse Matrices 102
4: Elementary Matrices: Rows Get Equal Time 110
5: The Transpose 119
Chapter 3: Vector Spaces 127
1: Subspaces of Rn 127
2: The Four Fundamental Subspaces 136
3: Linear Independence and Basis 143
4: Dimension and Its Consequences 157
5: A Graphic Example 170
6: Abstract Vector Spaces 176
Chapter 4: Projections and Linear Transformations 191
1: Inconsistent Systems and Projection 191
2: Orthogonal Bases 200
3: The Matrix of a Linear Transformation and the Change-of-Basis Formula 208
4: Linear Transformations on Abstract Vector Spaces 224
Chapter 5: Determinants 239
1: Properties of Determinants 239
2: Cofactors and Cramer’s Rule 245
3: Signed Area in R2 and Signed Volume in R3 255
Chapter 6: Eigenvalues and Eigenvectors 261
1: The Characteristic Polynomial 261
2: Diagonalizability 270
3: Application 277
4: The Spectral Theorem 286
Chapter 7: Further Topics 299
1: Complex Eigenvalues and Jordan Canonical Form 299
2: Computer Graphics and Geometry 314
3: Matrix Exponentials and Differential Equations 331
For Further Reading 349
More on Linear Algebra 349
Historical Matters 349
Other Interesting Sources 349
Answers to Selected Exercises 351
List of Blue Boxes 367
