Mathematics
Learn the core ideas of linear algebra through intuitive, visual explanations from 3Blue1Brown.
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Now playing: Vectors | Chapter 1, Essence of linear algebra
This learning path follows the popular Essence of linear algebra series by 3Blue1Brown. It covers the fundamental concepts of linear algebra, from vectors and matrices to eigenvalues and abstract vector spaces, using a visuals-first approach to build deep intuition. The path is organized into sections that group related videos, preserving the original order. Each video includes a summary and specific learning objectives to guide your study.
Beginners who want a structured free introduction before deeper practice.
Lesson 1: Vectors | Chapter 1, Essence of linear algebra
This video introduces the series and the basic concept of vectors. It explains vectors as arrows in space and as lists of numbers, and sets the stage for the visual approach used throughout the course.
Open on YouTubeLesson 2: Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
This video covers linear combinations of vectors, the span of a set of vectors, linear dependence, and the idea of basis vectors. It shows how these concepts form the foundation for understanding vector spaces.
Open on YouTubeLesson 3: Linear transformations and matrices | Chapter 3, Essence of linear algebra
This video introduces linear transformations and their representation as matrices. It emphasizes that a matrix is a way to describe a linear transformation, and shows how to visualize these transformations in 2D.
Open on YouTubeLesson 4: Matrix multiplication as composition | Chapter 4, Essence of linear algebra
This video explains matrix multiplication as the composition of two linear transformations. It shows that multiplying matrices corresponds to applying one transformation after another, and clarifies why the order matters.
Open on YouTubeLesson 5: Three-dimensional linear transformations | Chapter 5, Essence of linear algebra
This short video extends the visual intuition of linear transformations to three dimensions. It shows how 3D transformations can be represented by 3x3 matrices and how they affect the grid in 3D space.
Open on YouTubeLesson 6: The determinant | Chapter 6, Essence of linear algebra
This video introduces the determinant, a scalar that measures how much a linear transformation scales areas or volumes. It explains the geometric meaning of the determinant and how to compute it for 2x2 and 3x3 matrices.
Open on YouTubeLesson 7: Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra
This video covers inverse matrices, column space, and null space in the context of solving linear systems. It provides a geometric interpretation of these concepts and explains when a system has a unique solution, no solution, or infinitely many solutions.
Open on YouTubeLesson 8: Nonsquare matrices as transformations between dimensions | Chapter 8, Essence of linear algebra
This video briefly discusses non-square matrices, which represent transformations between spaces of different dimensions. It shows how such matrices map vectors from one dimension to another and how the concepts of rank and null space still apply.
Open on YouTubeLesson 9: Dot products and duality | Chapter 9, Essence of linear algebra
This video explains the dot product from both geometric and algebraic perspectives, and introduces the concept of duality. It shows why the formula for the dot product matches the geometric intuition of projection.
Open on YouTubeLesson 10: Cross products | Chapter 10, Essence of linear algebra
This video covers the geometric intuition behind the cross product in 2D and 3D. It explains how the cross product produces a vector perpendicular to two given vectors, with magnitude equal to the area of the parallelogram they span.
Open on YouTubeLesson 11: Cross products in the light of linear transformations | Chapter 11, Essence of linear algebra
This video provides a deeper understanding of the cross product by relating it to linear transformations and duality. It explains why the formula for the cross product works and how it connects to determinants.
Open on YouTubeLesson 12: Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra
This video explains Cramer's rule geometrically, showing why it works for solving linear systems. It provides a visual proof that connects the rule to areas and determinants.
Open on YouTubeLesson 13: Change of basis | Chapter 13, Essence of linear algebra
This video discusses change of basis, which is how to translate between coordinate systems that use different basis vectors. It explains the role of the change of basis matrix and how to use it to convert coordinates.
Open on YouTubeLesson 14: Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra
This video provides a visual understanding of eigenvectors and eigenvalues. It explains that eigenvectors are vectors that remain on their span during a transformation, and eigenvalues are the factors by which they are scaled. It also discusses the usefulness of an eigenbasis.
Open on YouTubeLesson 15: A quick trick for computing eigenvalues | Chapter 15, Essence of linear algebra
This video presents a quick trick for computing the eigenvalues of a 2x2 matrix by inspection. It shows how to find the trace and determinant to quickly determine the eigenvalues.
Open on YouTubeLesson 16: Abstract vector spaces | Chapter 16, Essence of linear algebra
This video introduces abstract vector spaces, explaining that the concepts of linear algebra apply to more than just geometric vectors. It discusses functions as vectors and other examples, showing why linear algebra is so powerful.
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