Linear Algebra

Linear Algebra Course Menu

This course includes the study of vectors in the plane and space, systems of linear equations, matrices, determinants, vector spaces, linear transformations, inner products, eigenvalues, eigenvectors, diagonalization, matrix decomposition, and the Spectral Decomposition theorem.
Prerequisite: Calculus l with a grade of C or better. Methods of Proof is recommended.

Chapter 1

Linear Equations in Linear Algebra

Lessons Homework
1.1 Systems of Linear Equations 1.1
1.2 Row Reduction and Echelon Forms 1.2
1.3 Vector Equations 1.3
1.4 The Matrix Equation Ax = b 1.4
1.5 Solution Sets of Linear Systems 1.5
1.6 Applications of Linear Systems 1.6
1.7 Linear Independence 1.7
1.8 Introduction to Linear Transformations 1.8
1.9 The Matrix of a Linear Transformation 1.9

Chapter 2

Matrix Algebra

Lessons Homework
2.1 Matrix Operations 2.1
2.2 The Inverse of a Matrix 2.2
2.3 Characterizations of Invertible Matrices 2.3
2.4 Partitioned Matrices 2.4
2.5 Matrix Factorizations 2.5
2.8 Subspaces of Rn 2.8
2.9 Dimension and Rank 2.9

Chapter 3

Determinants

Lessons Homework
3.1 Introduction to Determinants 3.1
3.2 Properties of Determinants 3.2
3.3 Cramer's Rule, Volume and Linear Transformations 3.3

Chapter 4

Vector Spaces

Lessons Homework
4.1 Vector Spaces and Subspaces 4.1
4.2 Null Spaces, Column Spaces, and Linear Transformations 4.2
4.3 Linearly Independent Sets; Bases 4.3
4.4 Coordinate System 4.4
4.5 The Dimension of a Vector Space 4.5
4.6 Rank 4.6
4.7 Change of Basis 4.7

Chapter 5

Eigenvalues and Eigenvectors

Lessons Homework
5.1 Eigenvalues and Eigenvectors 5.1
5.2 The Characteristic Equation 5.2
5.3 Diagonalization 5.3
5.4 Eigenvectors and Linear Transformations 5.4
5.5 Complex Eigenvalues 5.5

Chapter 6

Orthogonality and Least Squares

Lessons Homework
6.1 Inner Product, Length, and Orthogonality 6.1
6.2 Orthogonal Sets 6.2
6.3 Orthogonal Projections 6.3
6.4 The Gram-Schmidt Process 6.4
6.5 Least-Squares Problems 6.5
6.6 Applications to Linear Models 6.6
6.7 Inner Product Spaces 6.7

Chapter 7

Symmetric Matrices and Quadratic Forms

Lessons Homework
7.1 Diagonalization of Symmetric Matrices 7.1
7.2 Quadratic Forms 7.2
7.3 Constrained Optimization 7.3
7.4 The Singular Value Decomposition 7.4
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