1.1. Systems of Linear Equations

• Linear Equations
• Systems of Linear Equations
• Solutions to Systems of 2-Variables and 3-Variables
• Homogenous Linear Systems

1.2. Introduction to Matrices and Elementary Row Operations

• Matrices and Elementary Row Operations
• Row Echelon Form of a Matrix (REF)
• Gaussian Elimination and Back Substitution
• Reduced Row Echelon Form of a Matrix (RREF)
• Gauss-Jordan Elimination

1.3. Operations on Matrices, and Special Types of Matrices

• Matrix Addition and Subtraction
• Scalar Multiples of Matrices
• The Transpose of a Matrix
• The Trace of a Square Matrix
• Matrix Multiplication
• Zero Matrices
• Diagonal Matrices
• Triangular Matrices
• Symmetric Matrices
• Identity Matrices
• Inverse of a Matrix
• Inverse of Diagonal Matrices
• Powers of a Matrix
• Matrix Polynomials
• Elementary Matrices
• Finding a Matrix's Inverse with Elementary Matrices
• Finding a Matrix's Inverse with Row Operations

2.1. The Determinant of Square Matrices

• Evaluating 2x2 Determinants and Matrix Inverses
• Minor and Cofactor Entries
• Evaluating nxn Determinants with Minor and Cofactor Entries of a Matrix
• Determinants of Triangular and Diagonal Matrices
• Effects of Elementary Matrix Operations on Determinants
• Determinants of Elementary Matrices
• Evaluating Determinants by Row Reduction
• Combinatorial Approach to Determinants
• Properties of Determinants
• Adjoint Matrices
• Finding a Matrix Inverse with its Determinant and Adjoint
• Cramer's Rule for Systems of Linear Equations

3.1. Vectors in Euclidean Space

• Scalars and Vectors
• Euclidean Space
• Basic Properties of Vectors
• Determining a Vector Given Two Points
• Translation Equations
• The Norm of a Vector
• Unit Vectors
• Vector Dot Product (Euclidean Inner Product)
• Orthogonal (Perpendicular) Vectors
• Orthogonal Projections
• Vector Cross Product
• Lagrange's Identity
• Standard Unit Vectors

3.2. Vector Geometry in Euclidean Space

• The Areas of Parallelograms and Triangles in 3-Space
• The Area of a Parallelogram in 2-Space
• Scalar Triple Products
• The Volume of a Parallelepiped in 3-Space
• Point-Normal Form of a Plane
• Vector Form Equations of a Plane
• Parametric Equations of Lines
• Lines of Intersection Between Two Planes
• The Distance Between a Plane and a Point
• The Distance Between Parallel Planes
• The Distance Between Two Vectors

3.3. Transformations in Euclidean Space

• Contraction and Dilation Transformations
• Reflection Transformations
• Projection Transformations
• Rotational Transformations

4.1. Fields

• Complex Numbers ( Examples 1 )
• The Complex Conjugate of a Complex Number
• The Absolute Value of a Complex Number
• Field Axioms
• The Field of Complex Numbers
• The Field of Rational Functions
• Galois Fields

5.1. Introduction to Vector Spaces

• Vector Spaces
• The Vector Space of n-Component Vectors
• The Vector Space of m x n Matrices
• The Vector Space of Lines Through the Origin of R2
• The Zero Vector Space
• Further Examples of Vector Spaces
• Properties of Vector Spaces
• Determining Whether a Set is a Vector Space
• Vector Subspaces ( Examples 1 | Examples 2 )
• The Vector Subspace of 2 x 2 Matrices
• The Vector Subspace of Real-Valued Continuous Functions
• Vector Subspaces of Homogenous Systems of Rn
• Vector Subspace Sums ( Examples 1 | Examples 2 )
• Vector Sum Theorems
• Direct Sum Theorems
• The Intersection and Union of Subspaces
• Solution Spaces of Homogenous Linear Systems
• Vector Spaces Review

6.1. Linear Independence and Spanning Sets

• Linear Combinations of a Set of Vectors
• Span of a Set of Vectors
• Spanning Set of a Vector Space ( Examples 1 )
• Finite and Infinite-Dimensional Vector Spaces ( Examples 1 )
• Linear Independence and Dependence ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 )
• Theorems Regarding Linear Independence and Dependence
• Linear Dependence Lemma
• Finite-Dimensional Linearly Independent Set of Vectors Theorem
• Infinite/Finite-Dimensional Vector Space Comparison Theorem
• Infinite-Dimensional Vector Space Theorem
• Linear Independence and Spanning Sets Review

6.2. Bases and the Dimension of Vector Spaces

• Basis of a Vector Space ( Examples 1 | Examples 2 | Examples 3 )
• Theorems Regarding a Basis of a Vector Space
• Finding a Basis for a Set of Vectors
• Dimension of a Vector Space ( Examples 1 )
• Sufficient Conditions for a Set of Vectors to be a Basis
• The Dimension of a Sum of Subspaces ( Examples 1 | Examples 2 )
• The Dimension of a Direct Sum of Subspaces
• Bases and Dimension Review

7.1. Linear Maps

• Linear Maps ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )
• Linear Maps Defined by Bases
• Linear Maps from Fn to Fm
• Addition, Multiples, and Compositions of Linear Maps ( Examples 1 )
• Null Space of a Linear Map
• Range of a Linear Map ( Examples 1 )
• Injective and Surjective Linear Maps ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )
• The Dimension of The Null Space and Range ( Examples 1 | Examples 2 | Examples 3 )
• Linear Maps Review

7.2. Matrices of Linear Maps

• Solution Spaces to Systems of Linear Equations
• The Matrix of a Linear Map ( Examples 1 )
• Properties of The Matrix of a Linear Map
• Invertibility of a Linear Map ( Examples 1 | Examples 2 )
• Vector Space Isomorphisms ( Examples 1 )
• Linear Operators
• Matrices of Linear Maps Review

8.1. Polynomials

• Properties of Polynomials ( Examples 1 | Examples 2 )
• The Fundamental Theorem of Algebra
• The Factorization of Polynomials with Complex Coefficients
• Pairs of Complex Roots for Polynomials with Real Coefficients
• The Factorization of Polynomials with Real Coefficients
• Root-Finding Techniques
• Polynomials Review

9.1. Eigenvalues

• Invariant Subspaces ( Examples 1 | Examples 2 )
• Eigenvalues and Eigenvectors ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 | Examples 6 )
• Polynomials Applied to Linear Operators ( Examples 1 | Examples 2 )
• The Existence of an Eigenvalue on Finite-Dimensional Complex Vector Spaces
• Upper Triangular Matrices of Linear Operators
• Upper Triangular Matrices for Operators on Complex Vector Spaces
• Determining Eigenvalues from Upper Triangular Matrices of Linear Operators
• Diagonal Matrices of Linear Operators ( Examples 1 | Examples 2 | Examples 3 )
• Invariant Subspaces in Finite-Dimensional Real Vector Spaces
• Projection Operators
• The Existence of an Eigenvalue on an Odd-Dimensional Real Vector Space
• Eigenvalues Review

10.1. Inner Product Spaces

• Inner Product Spaces ( Examples 1 )
• Formulas for The Inner Product
• The Pythagorean Theorem for Inner Product Spaces ( Examples 1 )
• The Cauchy-Schwarz Inequality ( Examples 1 )
• The Triangle Inequality for Inner Product Spaces
• The Parallelogram Identity for Inner Product Spaces
• Inner Product Spaces Review

10.2. Orthonormal and Orthogonal Spaces

• Orthonormal Bases of Vector Spaces ( Examples 1 )
• The Gram-Schmidt Process ( Examples 1 )
• Orthogonal Complements ( Examples 1 )
• Orthogonal Projection Operators ( Examples 1 )
• Norm Minimization ( Examples 1
• Orthonormal Vectors Review

11.1. Linear Functionals and Adjoint Linear Operators

• Linear Functionals ( Examples 1 )
• The Adjoint of a Linear Map
• Properties of Adjoints of Linear Maps
• Determining the Adjoint of a Linear Map
• The Null Space and Range of the Adjoint of a Linear Map
• Injectivity and Surjectivity of the Adjoint of a Linear Map
• Eigenvalues of the Adjoint of a Linear Map
• The Conjugate Transpose of a Matrix
• The Matrix of the Adjoint of a Linear Map
• Self-Adjoint Linear Operators
• Eigenvalues of Self-Adjoint Linear Operators
• Self-Adjoint Linear Operators over Complex Vector Spaces
• Normal Linear Operators