If you're seeing this message, it means we're having trouble loading external resources for Khan Academy.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Skip to main content
Subjects
  • Math by subject
    • Early math
    • Arithmetic
    • Algebra
    • Geometry
    • Trigonometry
    • Statistics & probability
    • Calculus
    • Differential equations
    • Linear algebra
    • Math for fun and glory
  • Math by grade
    • K–2nd
    • 3rd
    • 4th
    • 5th
    • 6th
    • 7th
    • 8th
  • Science & engineering
    • Physics
    • Chemistry
    • Organic chemistry
    • Biology
    • Health & medicine
    • Electrical engineering
    • Cosmology & astronomy
  • Computing
    • Computer programming
    • Computer science
    • Hour of Code
    • Computer animation
  • Arts & humanities
    • Art history
    • Grammar
    • Music
    • US history
    • World history
  • Economics & finance
    • Microeconomics
    • Macroeconomics
    • Finance & capital markets
    • Entrepreneurship
  • Test prep
    • SAT
    • MCAT
    • GMAT
    • IIT JEE
    • NCLEX-RN
    • CAHSEE
  • College Admissions
    DonateSign in / Sign up
    Search for subjects, skills, and videos
    Main content
    Linear algebra

    Essence of linear algebra

    Contents
    About
    The geometry of vectors and matrices
    Looking for a quick review of the core topics in linear algebra? Or perhaps you want to preview the relevant intuitions before diving in deeper. This sequence is focussed on the geometric intuitions underlying the subject.

    The geometry of vectors and matrices

    For anyone looking to review/preview linear algebra, this sequence of videos by one of our content creators Grant Sanderson illustrates many of the geometric intuitions underlying the core concepts surrounding matrices and vectors.
    Essence of linear algebra preview
    Chapter 1: Vectors, what even are they?
    Chapter 2: Linear combinations, span, and bases
    Chapter 3: Linear transformations and matrices
    Chapter 4: Matrix multiplication as composition
    Footnote: Three-dimensional linear transformations
    Chapter 5: The determinant
    Chapter 6: Inverse matrices, column space and null space
    Footnote: Nonsquare matrices as transformations between dimensions
    Chapter 7: Dot products and duality
    Chapter 8: Cross products
    Chapter 8 part 2: Cross products in the light of linear transformations
    Chapter 9: Change of basis
    Chapter 10: Eigenvectors and eigenvalues
    Chapter 11: Abstract vector spaces
    Vectors and spaces
    Up Next
    Vectors and spaces