ABOUT

To the Instructor

How to Use this Text

Chapter 1: Preliminaries

1.1 Introduction to

1.2 Introduction to Vectors

1.3 Length of a Vector

1.4 Vector Arithmetic

1.5 Standard Unit Vectors in

1.6 Unit Vector in the Direction of a Given Vector

1.7 Dot Product and its Properties

1.8 Dot Product and Angle

1.9 Orthogonal Projections

1.10 Cross Product and its Properties

1.11 Parametric Equations of Lines

1.12 Equations of Planes

Chapter 2: Systems of Equations

2.1 Introduction to Systems of Linear Equations

2.2 Augmented Matrix Notation and Elem. Row Ops.

2.3 Gaussian Elimination and Rank

2.4 Iterative Methods for Solving Linear Systems

Additional Exercises for Chapter 2

Chapter 3: Big Ideas about Vectors

3.1 Linear Combinations of Vectors

3.2 Span

3.3 Linear Independence

Additional Exercises for Chapter 3

Chapter 4: Matrices

4.1 Matrix Addition and Scalar Multiplication

4.2 Matrix Multiplication

4.3 Block Matrix Multiplication

4.4 Transpose of a Matrix

4.5 Linear Systems as Matrix and Linear Combination Equations

4.6 Homogeneous and Nonhomogeneous Systems

4.7 The Inverse of a Matrix

4.8 Elementary Matrices

4.9 LU Factorization

Additional Exercises for Chapter 4

Chapter 5: Subspaces of

5.1 and Subspaces of

5.2 Introduction to Bases

5.3 Bases and Dimension

5.4 Null(A), col(A), row(A) and Rank-Nullity theorem

Additional Exercises for Chapter 5

Chapter 6: Linear Transformations of

6.1 Matrix Transformations

6.2 Geometric Transformations of the Plane

6.3 Introduction to Linear Transformations

6.4 Standard Matrix of a Linear Transformation from Rn to Rm

6.5 Composition and Inverses

6.6 Kernel and Image of a Linear Transformation

Additional Exercises for Chapter 6

Chapter 7: Determinants

7.1 Finding the Determinant

7.2 Determinants, Areas, and Volumes

7.3 Elementary Row Operations and the Determinant

7.4 Properties of Determinants

7.5 Tedious Proofs Concerning Determinants

7.6 Determinants and Inverses of Nonsingular Matrices

Additional Exercises for Chapter 7

Chapter 8: Eigenvalues

8.1 Describing Eigenvalues and Eigenvectors

8.2 The Characteristic Equation

8.3 Similar Matrices and their Properties

8.4 Diagonalizable Matrices/Multiplicity

8.5 Gershgorin’s Theorem

8.6 Power Method and the Dominant Eigenvalue

Additional Exercises for Chapter 8

Chapter 9: Orthogonality

9.1 Orthogonality and Projections

9.2 Gram-Schmidt Orthogonalization

9.3 Orthogonal Complements and Decompositions

9.4 Orthogonal Matrices and Symmetric Matrices

9.5 Positive Definite Matrices

9.6 QR Factorization

9.7 Least-Squares

9.8 SVD Decomposition

Additional Exercises for Chapter 9

Chapter 10: Abstract Vector Spaces

10.1 Abstract Vector Spaces

10.2 Bases and Dimension for Abstract Vector Spaces

10.3 Linear Transformations of Abstract Vector Spaces

10.4 Existence of Inverses of Linear Transformations

10.5 Isomorphic Vector Spaces

10.6 Matrices of Linear Transformations with Respect to Arbitrary Bases

10.7 Inner Product Spaces

Additional Exercises for Chapter 10

Chapter 11: Applications

11.1 Application to Network Flow

11.2 Application to Electrical Networks

11.3 Application to Chemical Equations

11.4 Application to Input-Output Economic Models

11.5 Application to Markov Chains

11.6 Application to Computer Graphics

Chapter 12: Appendix

12.1 The Triangle Inequality

12.2 Complex Numbers

12.3 Complex Matrices

12.4 INDEX