Math 511: Linear Algebra

Table of Contents

Light Mode ¶

Chapter 1 - Linear Systems of Equations¶

1.1 Systems of Linear Equations

1.2 Matrix Equations

1.3 Applications

P1 Graphing Linear Equations

P2 Underdetermined and Overdetermined Systems

Chapter 2 - Matrices¶

2.1 Matrix Arithmetic

2.2 Matrix Algebra

2.3 Matrix Inverse

2.4 Elementary Matrices

2.5 Markov Processes

2.6 Applications

P3 Exploring Multiplication

P4 Nilpotent Matrices

Chapter 3 - Determinants¶

3.1 Determinants

3.2 The Laplace Expansion

3.3 Applications of Determinant

3.4 Review for Test 1

P5 Stochastic Matrices

P6 The Cayley-Hamilton Theorem

Chapter 4 - Vector Spaces¶

4.1 Vectors in $\mathbb{R}^n$

4.2 Vector Spaces

4.4 Span and Linear Independence

4.5 Basis and Dimension

4.6 The Four Fundamental Subspaces

4.7 Changing Coordinates

4.8 Applications

P7 Solutions of Linear Systems

Chapter 5 - Inner Product Spaces¶

5.1 Geometry in Euclidean Vector Spaces

5.2 Inner Product Spaces

5.3 Orthonormal Bases

5.3b Gram-Schmidt Orthogonalization

5.4 Least Squares

5.5 Applications

5.6 Orthogonal Subspaces

5.7 Review for Test 2

P9 QR Factorization

P10 Orthogonal Matrices and Change of Basis

Chapter 6 - Linear Transformations¶

6.1 Definitions and Examples

6.2 Kernel and Range

6.3 Linear Transformations as Matrices

6.4 Similar Matrices

6.5 Applications

P11 Reflections on the Plane (Part 1)

P12 Reflections on the Plane (Part 2)

Chapter 7 - Eigenvalues¶

7.1 Eigenvalues and Eigenvectors

7.2 Diagonalization

7.3 Hermitian Matrices

7.5 Singular Value Decomposition

P13 Singular Value Decomposition

Additional Links and Resources¶

3Blue1Brown - The Essence of Linear Algebra

A wonderful series of short videos by Grant Sanderson that explore the fundamental ideas of linear algebra in a way that provides visualization and context for the primary concepts. They serve as both a gentle introduction to some of the tougher concepts and a great way to review and reinforce the main ideas.

A Vision of Linear Algebra

MIT provides many online resources through their Open Courseware program. Linked here is a full class, including video lectures, featuring MIT professor Gilbert Strang. The authors of these notes take inspiration from his lectures on this subject for our own writing here and classroom presentations.