Linear Algebra

Matrices, vectors, and more - from theory to the real world!

Linear algebra is pervasive in just about all modern scientific subjects, including physics, mathematics, computer science, electrical engineering, economics, and aeronautical engineering. You’ll learn about its applications in computer graphics, signal processing, machine learning, RLC circuit analysis, and control theory.

By the end of this course, you’ll be able to solve systems of equations of all flavors and complexities using linear algebra, from a simple 2x2 matrix equation to much more complex systems involving many variables.

Interactive
quizzes

26

Concepts and
exercises

255+
  1. 1

    Linear Equations

    Multiple variables, multiple equations - no worries!

    1. Two Linear Equations in Two Unknowns

      Kick things off with a pair of equations in a pair of unknowns.

    2. Three Unknowns

      Increase the challenge with three equations in three unknowns.

    3. Gaussian Elimination

      Learn a general algorithm for solving systems of equations.

    4. The Full Story of Gaussian Elimination

      Look deeper into the math behind Gauss-Jordan reduction.

  2. 2

    Vector Spaces

    Explore the power of vectors with magic squares, spanning sets, and more...

    1. Included with
      Brilliant Premium

      Vector Spaces

      What is the essence of a vector?

    2. Included with
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      Subspaces and Span

      Discover what it means to have a space within a space.

    3. Included with
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      Linear Independence

      Learn how to spot redundant vectors.

    4. Included with
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      Basis and Dimension

      What measures the size of a vector space?

  3. 3

    Properties of Matrices

    A fundamental building block for linear algebra.

    1. Included with
      Brilliant Premium

      Matrix Algebra

      Find out what matrices and vectors share in common.

    2. Included with
      Brilliant Premium

      Inverses and Systems of Equations

      Is it ever OK to divide by a matrix?

    3. Included with
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      Four Fundamental Subspaces

      See what the column, row, null and transpose kernel spaces have to say about a matrix.

    4. Included with
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      Adjacency Matrices

      Practice fundamental matrix concepts on a graph theory application.

  4. 4

    Linear Maps and Matrices

    Determinants, maps, bases, and more.

    1. Included with
      Brilliant Premium

      Linear Transformations

      Learn how to turn vectors into... other vectors.

    2. Included with
      Brilliant Premium

      Properties of Linear Transformations

      Uncover the deep connection between linear transformations and matrices.

    3. Included with
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      2x2 Determinants

      Discover a simple test for matrix invertibility.

    4. Included with
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      Determinants in Higher Dimensions

      Explore some beautiful and useful properties of determinants.

  5. 5

    Eigenvalues and Diagonalizability

    Eigenvalues, eigenvectors, and applications!

    1. Included with
      Brilliant Premium

      Eigenvalues and Eigenvectors

      Learn to work with special vectors of fundamental importance.

    2. Included with
      Brilliant Premium

      Characteristic Polynomial

      Learn a surefire way to find eigenvalues.

    3. Included with
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      Diagonalizability

      When can a matrix be put into a diagonal form?

    4. Included with
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      PageRank and Exponentiation

      Sample some of the uses of diagonalization in graph theory and probability.