Essence of Linear Algebra: Free Course

About this course

See vectors and matrices as meaningful geometric objects in this free Essence of Linear Algebra course from 3Blue1Brown. All sixteen numbered chapters are preserved, from vectors and span to transformations, determinants, eigenvectors and abstract vector spaces. The sequence develops a connected visual foundation for later mathematical and technical study.

Course at a glance

What you’ll learn

• Connect vectors, linear combinations, span and basis with geometric meaning.
• Interpret matrices and their multiplication as linear transformations.
• Explore determinants, inverse matrices, dot products, cross products and changes of basis.
• Understand the visual purpose of eigenvectors, eigenvalues and abstract vector spaces.

Skills you’ll gain

• Vectors
• Matrix transformations
• Determinants
• Change of basis
• Eigenvectors
• Geometric reasoning

Course Overview

The first chapters establish vectors, linear combinations and basis before the course introduces matrices as transformations. Matrix multiplication then becomes a way of composing those transformations rather than an isolated table of arithmetic. Later chapters expand into determinants, inverse matrices, column and null spaces, dot and cross products, Cramer’s rule and changes of basis. The final portion addresses eigenvectors, eigenvalues and abstract vector spaces. All sixteen chapters remain in their numbered order. The series emphasizes intuition, so it works particularly well alongside independent calculation practice. It does not replace every proof, exercise or assessment in a formal linear-algebra class, but gives you a stronger foundation for understanding why the calculations have the forms they do.

How to study the course

• Follow Chapters 1–5 and sketch the effect of a simple transformation on a vector and a grid. Explain the picture before calculating a more complicated example.
• Study Chapters 6–13 and connect each new operation with a geometric question. Revisit span and basis whenever the terminology becomes hard to distinguish.
• Complete Chapters 14–16, including the eigenvalue trick and abstract-vector-space discussion. Keep the numbered sequence intact when reviewing a concept that depends on an earlier chapter.

Use the searchable classroom lesson list to revisit a specific explanation. Pause after a useful demonstration, reproduce a small example and explain the result in your own words. Keep a short note of what worked, what changed and what you still need to understand before moving on.

Who is this for?

This course suits students beginning or revising linear algebra, programmers studying mathematical foundations and learners who prefer visual explanations. It is useful when you can perform a matrix calculation but want to understand what the result represents.

Before you start

Comfort with algebra, coordinates and basic functions is helpful. A notebook is enough for visual study; independent numerical examples add useful practice. No paid software is required for viewing, and the classroom does not provide graded university exercises or academic credit.

Put the lessons into practice

Choose a small two-dimensional matrix and describe how it changes the basis directions. Draw a grid and a few vectors before and after the transformation, then compare the picture with the arithmetic. Create a second example that changes area differently and explain the role of the determinant. Finally, find a simple direction whose behavior helps you discuss an eigenvector. Keep the examples small enough to verify and write a short explanation for someone who has not seen the course. Return to the relevant chapter whenever the calculation and picture disagree.

This optional activity is provided by WikiFree for independent practice. It is not a graded assignment from the instructor. Keep your working notes, check the result against the task you defined and return to the relevant lesson when an assumption or technique is unclear.

Certificate / Assessment

Watching the public videos and using WikiFree progress tracking does not award a certificate, exam voucher, academic credit or instructor assessment.

Use your practice output to review what you can now explain and repeat independently. Completion tracking records your study progress; it does not replace an assessed qualification or professional experience.

Frequently asked questions

Provider: 3Blue1Brown. Videos play from the 3Blue1Brown channel on YouTube.