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