Mathematics, Statistics & Physics

Math 511: Linear Algebra¶

Table of Contents¶

Chapter 1 - Linear Systems of Equations¶

1.1 Systems of Linear Equations

1.2 Matrix Equations

1.3 Applications

Problem Set 1.1 - Graphing Linear Equations

Problem Set 1.2 - Underdetermined and Overdetermined Systems

Problem Set 1.3 - Linear Systems with MATLAB

Problem Set 1.4 - Formalism And Proof

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

Problem Set - 2.2 Exploring Multiplication

Problem Set - 2.1 Nilpotent Matrices

Chapter 3 - Determinants¶

3.1 Determinants

3.2 The Laplace Expansion

3.3 Applications of Determinant

3.4 Review for Test 1

Problem Set - 3.1 Stochastic Matrices

Problem Set - 3.2 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

Problem Set 4.1 - Solutions of Linear Systems

Problem Set 4.2 - Direct Sum

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

Problem Set 5.1 - QR Factorization

Problem Set 5.2 - 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

Problem Set 6.1 - Reflections on the Plane (Part 1)

Problem Set 6.2 - 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

Problem Set 7.1 - Singular Value Decomposition

Problem Set 7.2 - True/False

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.