Free online courseUndergraduate Linear Algebra Full Course
Duration of the online course: 29 hours and 54 minutes
New
Free linear algebra course covering vectors, matrices, linear systems, determinants, eigenvalues, vector spaces, bases, and linear transformations.
In this free course, learn about
Course Overview and Vectors
Matrices and Linear Systems
Determinants and Eigenvalues
Vector Spaces: Subspaces, Span, Independence, and Bases
Coordinates, Linear Transformations, and Change of Basis
Dimension Theorem and Isomorphisms
Course Description
Build a solid undergraduate foundation in linear algebra with a free online course designed to take you from core vector skills to the big ideas that power modern math, science, and engineering. You will start with essential vector operations and the dot product, then move into matrix fundamentals and matrix multiplication, learning how these tools model and simplify real problems.
As the course progresses, you will develop practical methods for solving linear systems using Gaussian elimination and Gauss-Jordan reduction, including reduced row echelon form. You will also explore equivalent systems, rank, and row space, strengthening your ability to interpret what a system of equations is really saying and how matrix structure reveals solution behavior.
You will then deepen your understanding with inverses and determinants, including properties and computational techniques connected to row reduction. From there, the course transitions into the conceptual heart of linear algebra: eigenvalues and the language of vector spaces. You will study subspaces, span, linear independence, and basis, including strategies for constructing special bases and working confidently with coordinates.
Finally, you will connect the algebra to functions through linear transformations, learn how to represent transformations with matrices, and tie key ideas together with the dimension theorem and isomorphisms. This course fits learners who want a comprehensive, step-by-step path through algebra topics commonly covered in university linear algebra.
Course content
Video class: Course Introduction - Linear Algebra - Lecture 0 (of 23)09m
Exercise: Which set of broad sections best matches the course outline?
Video class: Fundamental Operations with Vectors - Linear Algebra - Lecture 1 (of 23)1h02m
Exercise: Which statement correctly defines when two vectors are in the same direction?
Video class: The Dot Product - Linear Algebra - Lecture 2a (of 23)54m
Exercise: Which formula correctly defines the dot product of vectors x and y in R^n?
Video class: The Dot Product - Linear Algebra - Lecture 2b (of 23)45m
Exercise: Which formula gives the projection of vector b onto a nonzero vector a in \(\mathbb{R}^n\)?
Video class: Fundamental Operations with Matrices - Linear Algebra - Lecture 3 (of 23)1h17m
Exercise: When is the sum A + B of two matrices defined?
Video class: Matrix Multiplication - Linear Algebra - Lecture 4a (of 23)58m
Exercise: If A is an M×N matrix and B is an N×P matrix, what are the dimensions of the product AB (assuming it is defined)?
Video class: Matrix Multiplication - Linear Algebra - Lecture 4b (of 23)30m
Exercise: Which statement is true about the cancellation law for matrix multiplication?
Video class: Solving Linear Systems Using Gaussian Elimination - Linear Algebra - Lecture 5a (of 23)1h06m
Exercise: Which set lists all possible sizes of the solution set for a linear system?
Video class: Solving Linear Systems Using Gaussian Elimination - Linear Algebra - Lecture 5b (of 23)1h11m
Exercise: In Gaussian elimination, which row operation is primarily used to create zeros below a pivot?
Video class: Gauss-Jordan and Reduced Row Echelon Form (RREF) - Linear Algebra - Lecture 6 (of 23)56m
Exercise: Which condition guarantees that a linear system has no solution when its augmented matrix is in reduced row echelon form (RREF)?
Video class: Equivalent Systems, Rank, and Row Space - Linear Algebra - Lecture 7a (of 23)57m
Exercise: What does it mean for two systems of linear equations to be equivalent?
Video class: Equivalent Systems, Rank, and Row Space - Linear Algebra - Lecture 7b (of 23)28m
Exercise: How can you check whether a vector x is in the row space of a matrix A?
Video class: Inverses of Matrices - Linear Algebra - Lecture 8 (of 231h09m
Exercise: How can you find the inverse of a general n×n matrix A (n>2) using row operations?
Video class: Introduction to Determinants - Linear Algebra - Lecture 9 (of 23)57m
Exercise: In the 3×3 basket weaving method for determinants, which diagonals are added with a positive sign?
Video class: Determinants and Row Reduction - Linear Algebra - Lecture 10 (of 23)46m
Exercise: How does the determinant change when two rows of a matrix are swapped?
Video class: Additional Properties of Determinants - Linear Algebra - Lecture 11 (of 23)1h09m
Exercise: Which statement correctly describes the determinant of a product of two n×n matrices?
Video class: Eigenvalues55m
Exercise: How can you determine whether a real number \(\lambda\) is an eigenvalue of an \(n\times n\) matrix \(A\)?
Video class: Eigenvalues42m
Exercise: Which condition guarantees that an $n\times n$ matrix $A$ is diagonalizable?
Video class: Introduction of Vector Spaces - Linear Algebra - Lecture 13 (of 23)1h08m
Exercise: Which condition is necessary for the set \(P_n\) to form a vector space under polynomial addition and scalar multiplication?
Video class: Subspaces - Linear Algebra - Lecture 14 (of 23)58m
Exercise: Which condition is sufficient to prove that a non-empty subset W of a vector space V is a subspace of V (using the same operations as V)?
Video class: Span - Linear Algebra - Lecture 15a (of 23)54m
Exercise: Which statement best defines the span of a non-empty subset S of a vector space V?
Video class: Span - Linear Algebra - Lecture 15b (of 23)34m
Exercise: How can you find a simplified set of vectors that spans the same subspace as a finite set S ⊂ ℝⁿ?
Video class: Linear Independence - Linear Algebra - Lecture 16a (of 23)54m
Exercise: Which condition guarantees that a finite non-empty set S = {v1, ..., vn} is linearly independent?
Video class: Linear Independence - Linear Algebra - Lecture 16b (of 23)27m
Exercise: Which statement correctly characterizes when a (possibly infinite) subset S of a vector space V is linearly independent?
Video class: Basis59m
Exercise: Which pair of conditions must a subset B satisfy to be a basis for a vector space V?
Video class: Basis25m
Exercise: If S is a finite subset of a finite-dimensional vector space V and S spans V, what must be true?
Video class: Constructing Special Bases - Linear Algebra - Lecture 18 (of 23)43m
Exercise: In the simplified span method for finding a basis of span(S), how are the vectors from S placed into the test matrix, and how is the basis read off after row reduction?
Video class: Coordinatization - Linear Algebra - Lecture 19a (of 23)58m
Exercise: What is the coordinatization of a vector w with respect to an ordered basis B = (v1, v2, ..., vn) if w = a1v1 + a2v2 + ... + anvn?
Video class: Coordinatization - Linear Algebra - Lecture 19b (of 23)36m
Exercise: What is the key property that defines the transition matrix from an ordered basis B to an ordered basis C?
Video class: Introduction to Linear Transformations - Linear Algebra - Lecture 20 (of 23)1h06m
Exercise: Which pair of properties must a function f: V → W satisfy to be a linear transformation?
Video class: The Matrix of a Linear Transformation - Linear Algebra - Lecture 21a (of 23)1h04m
Exercise: How is the matrix of a linear transformation (with respect to ordered bases B in V and C in W) constructed?
Video class: The Matrix of a Linear Transformation - Linear Algebra - Lecture 21b (of 23)22m
Exercise: How do you compute the matrix of a linear transformation after changing bases in the domain and codomain?
Video class: The Dimension Theorem - Linear Algebra - Lecture 22a (of 23)49m
Exercise: Which statement correctly expresses the Dimension Theorem for a linear transformation L: V → W with V finite-dimensional?
Video class: The Dimension Theorem - Linear Algebra - Lecture 22b (of 23)47m
Video class: Isomorphism - Linear Algebra - Lecture 23 (of 23)1h06m
Exercise: Which condition is sufficient to conclude a linear transformation L is one-to-one (injective)?
