Linear Algebra

Welcome to Linear Algebra. This course will cover Linear Equations, Matrix Algebra, Determinants, Vector Spaces, Eigenvalues and Eigenvectors, Orthogonality, and more! If you have any suggestions or would like more practice on a certain topic, please send your suggestions to


Linear Equations

Systems of Equations and Matrix Notation
Solving Systems of Equations
Solving Systems of Equations (Practice Questions)
Echelon Forms and Pivots
Solution Sets for Systems of Equations
Adding and Multiplying Vectors
Properties of Vectors
Linear Combinations
Span of Vectors
Matrix Equation Ax=b
Computing Ax and Identity Matrices
Proving Properties of Ax
Homogeneous Linear Systems and Parametric Form
Nonhomogeneous Linear Systems
Linear Independence
Linear Transformations
Matrices and Transformations
Geometric Transformations
Injective and Surjective Transformations
Linear Systems Exam Solutions

Matrix Algebra

Adding and Scaling Matrices
Multiplying Matrices
Matrix Transpose and Properties
Inverse Matrices and 2×2 Inverses
Solving Systems with Inverses, and Inverse Identities
Elementary Matrices and Invertibility

Vector Spaces

Vector Spaces
Null Space
Column Space
Kernel and Range
Basis and Linear Independence
Spanning Set Theorem
Coordinates and Change-of-Coordinates Matrix
Coordinate Mapping and Polynomials
Row Space and Rank Theorem
Change of Basis


Computing Determinants
Cofactor Expansion
Properties of Determinants

Eigenvectors and Eigenvalues

Eigenvalues and Eigenvectors
Eigenspace and Linear Independence
The Characteristic Equation

Orthogonality and Least-Squares


Vector Spaces

Subspace Proofs
Subspace Proofs 2
Vector Space Linear Mapping Proofs

Exams and Worksheets

Linear Algebra – Linear Systems Exam
Linear Algebra – Linear Systems Exam Solutions
Linear Algebra – Matrix Algebra Quiz
Linear Algebra – Matrix Algebra Quiz Solutions