Interactive course — Advanced Math

Linear Algebra

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

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Overview

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.

Topics covered

  • Bases
  • Determinants
  • Diagonalizable Matrices
  • Dot products
  • Eigenvalues and Eigenvectors
  • Gaussian Elimination
  • Inverses
  • Linear Independence
  • Linear Transformations
  • Matrices
  • Subspaces
  • Vector Spaces

Interactive quizzes

26

Concepts and exercises

255+

Course map

Prerequisites and Next Steps

  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.

      1
    2. Three Unknowns

      Increase the challenge with three equations in three unknowns.

      2
    3. Gaussian Elimination

      Learn a general algorithm for solving systems of equations.

      3
    4. The Full Story of Gaussian Elimination

      Look deeper into the math behind Gauss-Jordan reduction.

      4
    5. Application: Kirchhoff and Circuits

      Practice solving linear systems with electrical engineering problems.

      5
  2. 2

    Vector Spaces

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

    1. Vector Spaces

      What is the essence of a vector?

      6
    2. Subspaces and Span

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

      7
    3. Linear Independence

      Learn how to spot redundant vectors.

      8
    4. Basis and Dimension

      What measures the size of a vector space?

      9
    5. Dot Products and Inner Products

      What's the connection between vectors and geometry?

      10
    6. Least Squares

      Apply your linear algebra knowledge to an important problem in statistics.

      11
  3. 3

    Properties of Matrices

    A fundamental building block for linear algebra.

    1. Matrix Algebra

      Find out what matrices and vectors share in common.

      12
    2. Inverses and Systems of Equations

      Is it ever OK to divide by a matrix?

      13
    3. Four Fundamental Subspaces

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

      14
    4. Adjacency Matrices

      Practice fundamental matrix concepts on a graph theory application.

      15
  4. 4

    Linear Maps and Matrices

    Determinants, maps, bases, and more.

    1. Linear Transformations

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

      16
    2. Properties of Linear Transformations

      Uncover the deep connection between linear transformations and matrices.

      17
    3. 2x2 Determinants

      Discover a simple test for matrix invertibility.

      18
    4. Determinants in Higher Dimensions

      Explore some beautiful and useful properties of determinants.

      19
    5. Representation by a Matrix

      Formalize the relationship between matrices and linear transformations.

      20
    6. Change of Basis

      What makes two matrices similar?

      21
    7. Polynomial Interpolation

      Use determinants to fit a polynomial to a collection of data.

      22
  5. 5

    Eigenvalues and Diagonalizability

    Eigenvalues, eigenvectors, and applications!

    1. Eigenvalues and Eigenvectors

      Learn to work with special vectors of fundamental importance.

      23
    2. Characteristic Polynomial

      Learn a surefire way to find eigenvalues.

      24
    3. Diagonalizability

      When can a matrix be put into a diagonal form?

      25
    4. PageRank and Exponentiation

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

      26