1.1 Introduction to

1.6 Unit Vector in the Direction of a Given Vector

1.7 Dot Product and its Properties

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

Additional Exercises for Chapter 3

4.1 Matrix Addition and Scalar Multiplication

4.3 Block Matrix Multiplication

4.5 Linear Systems as Matrix and Linear Combination Equations

4.6 Homogeneous and Nonhomogeneous Systems

4.9 LU Factorization

Additional Exercises for Chapter 4

5.1 and Subspaces of

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

Additional Exercises for Chapter 5

**Chapter 6: Linear Transformations of **

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

6.6 Kernel and Image of a Linear Transformation

Additional Exercises for Chapter 6

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

8.1 Describing Eigenvalues and Eigenvectors

8.2 The Characteristic Equation

8.3 Similar Matrices and their Properties

8.4 Diagonalizable Matrices/Multiplicity

8.6 Power Method and the Dominant Eigenvalue

Additional Exercises for Chapter 8

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

Additional Exercises for Chapter 9

**Chapter 10: 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

10.6 Matrices of Linear Transformations with Respect to Arbitrary Bases

10.7 Inner Product Spaces

Additional Exercises for Chapter 10

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

12.2 Complex Numbers

12.3 Complex Matrices

12.4 INDEX