A Linear Algebra Course

Linear Algebra for Beginners

An intuitive and visual journey through linear algebra.

9 Modules
43 Lessons
Visual & Intuitive
Self-Paced

Course Contents

Work through the modules in order, or jump directly to a topic you're interested in.

Module 1

Linear Equations

The fundamental problem Ax = b and three ways to see it.

01
The Geometry of Linear Equations The row picture, column picture, and matrix picture.
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02
Elimination How systematic elimination solves a system of equations.
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03
Elimination Matrices Seeing elimination as multiplication by matrices.
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04
Solving Ax = b When a system has zero, one, or infinitely many solutions.
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Module 2

Matrix Algebra

The algebra of matrices and the structure behind linear systems.

05
Matrix Multiplication Why matrix multiplication works the way it does.
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06
Inverse Matrices Understanding the inverse of a matrix.
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07
A⁻¹ and Ax = b Connecting inverse matrices to solutions.
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08
LU Factorization Factoring a matrix into lower and upper triangular matrices.
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Module 3

Vector Spaces

Moving from individual vectors to the spaces they generate.

09
Vectors in Rⁿ Vectors, components, magnitude, and direction.
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10
Linear Combinations and Span How vectors combine and the space they can generate.
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11
Linear Independence When vectors provide genuinely new directions.
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12
Vector Spaces The structure that unifies vectors and their subspaces.
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13
Column Space The set of vectors that can be produced by a matrix.
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14
Nullspace Understanding all solutions to Ax = 0.
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Module 4

Basis & Dimension

Using independence and span to describe the structure of vector spaces.

15
Special Solutions Finding the fundamental solutions that build the nullspace.
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16
Basis A linearly independent set that spans a space.
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17
Dimension Counting the independent directions in a space.
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18
The Four Fundamental Subspaces Column space, nullspace, row space, and left nullspace.
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19
Rank Understanding the number of independent directions in a matrix.
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Module 5

Orthogonality & Least Squares

Geometry, projections, approximation, and orthogonal bases.

20
Orthogonal Vectors Perpendicularity through the dot product.
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21
Projections Finding the closest point on a line or subspace.
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22
Least Squares Solving systems that have no exact solution.
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23
Gram–Schmidt Turning independent vectors into an orthogonal basis.
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24
QR Factorization Matrix factorization built from orthogonal vectors.
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Module 6

Determinants

A number that captures scaling, orientation, and invertibility.

25
What Is a Determinant? Area, volume, and the geometry of a matrix.
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26
Properties of Determinants How row operations change determinants.
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27
Cofactors Expanding determinants and understanding their structure.
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28
Determinants and Volume Determinants as geometric scaling factors.
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Module 7

Eigenvalues & Eigenvectors

The special directions that reveal the internal structure of a matrix.

29
Eigenvalues and Eigenvectors Directions that a matrix does not rotate.
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30
Diagonalization Simplifying matrices using eigenvectors.
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31
Powers of a Matrix Using eigenvalues to understand repeated matrix operations.
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32
Symmetric Matrices Orthogonal eigenvectors and symmetric structure.
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33
Positive Definite Matrices Quadratic forms, energy, and stability.
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Module 8

Linear Transformations

Matrices as transformations and changes of coordinates.

34
Linear Transformations Thinking of matrices as functions between vector spaces.
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35
Matrices of Linear Transformations Representing transformations using matrices.
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36
Change of Basis Describing the same vector in different coordinate systems.
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37
Similar Matrices Different matrix representations of the same transformation.
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38
Singular Value Decomposition A powerful decomposition of any matrix.
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Module 9

Applications

Seeing linear algebra at work in real mathematical problems.

38
Graphs and Networks Using matrices to represent networks.
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39
Markov Matrices Linear algebra and long-run probabilities.
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40
Differential Equations and Eigenvalues Using eigenvalues to understand dynamic systems.
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41
Fourier Ideas Decomposing complicated signals into simpler components.
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42
Putting It All Together Connecting the major ideas of linear algebra.
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Ready to begin? Start with the geometry of linear equations.
Start Lesson 1 →