Table of Contents

Chapter 1

Introduction to Matrices and Linear Systems

The first chapter serves as an introduction to linear algebra. It starts with vectors defining matrices and their operations and then continues on to linear systems (which is a good starting point for most learners who are already familiar with them). Lastly, we will include determinants and invertibility as the last topic of the unit as we have sufficient knowledge to introduce these concepts, and I feel that they can be learned earlier in the course.

1.1 Linear Geometry: Vectors in Space and Angle & Length

• Definition of vector
• Dot product
• Angle between vectors
• Vector length

1.2 Matrix Definitions and Operations

• Matrix addition
• Matrix scalar multiplication
• Matrix multiplication

1.3 Solving Linear Systems

• Gauss’s method for solving linear systems, describing solution sets (particular + homogeneous)

1.4 Row Echelon Form and Reduce Row Echelon Form

• Gauss-Jordan Reduction and Linear Combination Lemma

1.5 Determinants and Invertibility

• Finding determinants
• Co-factor expansion
• Finding inverse matrices

Chapter 2

Deep Dive into Linear Algebra

In the second unit of the book, we are diving deeper into more complex topics of linear algebra. We first start with vector spaces and bases, which are one of the most important ideas. We then end the unit with projection, eigenvectors, and the fundamental theorem of linear algebra, which is built on all of the previous topics and has tremendous applications in coding and other fields.

2.1 Definition of Vector Spaces

• Subspaces and spanning sets

2.2 Linear Independence, Basis, and Dimension

• Finding the basis of a subspace and dimension of a matrix, identifying row space, column space, and null space

2.3 Use of Lambda (\(\lambda^{n}\))

• Eigenvalue, eigenvector, and eigenspace

2.4 Projection and Change of Basis

• Orthogonality, orthonormal basis, and Gram-Schmidt Process

2.5 Fundamental Theorem of Linear Algebra

Chapter 3

Beyond Linear Algebra