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Linear Algebra with Applications

Abstract vector spaces in theory and application.

Linear algebra plays a crucial role in many branches of applied science and pure mathematics. This course covers the core ideas of linear algebra and provides a solid foundation for future learning.

Using geometric intuition as a starting point, the course journeys into the abstract aspects of linear algebra that make it so widely applicable. By the end you'll know about vector spaces, linear transformations, determinants, eigenvalues & eigenvectors, tensor & wedge products, and much more.

Interactive
quizzes

41

Concepts and
exercises

490+
  1. 1

    Introduction to Vector Spaces

    1. What is a Vector?

      Discover the true nature of vectors.

    2. Waves as Abstract Vectors

      Take a visual tour of vector spaces.

    3. Why Vector Spaces?

      Experience the power of abstraction.

  2. 2

    System of Equations

    1. Included with
      Brilliant Premium

      The Gauss-Jordan Process I

      Gain experience with systems of equations through traffic planning.

    2. Included with
      Brilliant Premium

      The Gauss-Jordan Process II

      Learn a surefire method for cracking any set of linear equations.

    3. Included with
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      Application: Markov Chains I

      Apply your Gauss-Jordan skills to a classic probability problem.

  3. 3

    Vector Spaces

    1. Included with
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      Real Euclidean Space I

      Learn about important abstract concepts in a familiar setting.

    2. Included with
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      Real Euclidean Space II

      Lay the foundation for building vector spaces.

    3. Included with
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      Span & Subspaces

      Develop a quick means for generating vector spaces.

    4. Included with
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      Coordinates & Bases

      Condense common vector spaces with bases.

  4. 4

    Linear Transformations

    1. Included with
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      What Is a Matrix?

      Free your mind from viewing matrices as just arrays of numbers.

    2. Included with
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      Linear Transformations

      Come full circle and connect linear maps back to matrices.

    3. Included with
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      Matrix Products

      Find out one way of multiplying matrices together.

    4. Included with
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      Matrix Inverses

      Learn when it's OK to divide by a matrix.

  5. 5

    Multilinear Maps & Determinants

    1. Included with
      Brilliant Premium

      Bivectors

      Take the first step towards determinants with bivectors.

    2. Included with
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      Trivectors & Determinants

      Evaluate determinants like a pro with trivectors.

    3. Included with
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      Determinant Properties

      Gain experience with the most important properties of determinants.

    4. Included with
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      Multivector Geometry

      Learn about the visual aspects of multivectors.

  6. 6

    Eigenvalues & Eigenvectors

    1. Included with
      Brilliant Premium

      Application: Markov Chains II

      Discover eigenvectors by rethinking a classic probability problem.

    2. Included with
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      Eigenvalues & Eigenvectors

      Learn the essentials of eigenvalues & eigenvectors.

    3. Included with
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      Diagonalizability

      Restructure square matrices in the most useful way imaginable.

    4. Included with
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      Normal Matrices

      When can a matrix be diagonalized?

  7. 7

    Inner Product Spaces

    1. Included with
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      Inner Product Spaces

      Extend familiar geometric tools to abstract spaces.

    2. Included with
      Brilliant Premium

      Gram-Schmidt Process

      Practice making your very own orthonormal bases.

    3. Included with
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      Least Squares Regression

      Solve a crucial problem in statistics with inner product spaces.

    4. Included with
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      Singular Values & Vectors

      Build singular values & vectors with least squares regression.