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Brilliant

Introduction to Linear Algebra

via Brilliant

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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.

Syllabus

  • Linear Equations: Multiple variables, multiple equations - no worries!
    • Two Linear Equations in Two Unknowns: Kick things off with a pair of equations in a pair of unknowns.
    • Three Unknowns: Increase the challenge with three equations in three unknowns.
    • Gaussian Elimination: Learn a general algorithm for solving systems of equations.
    • The Full Story of Gaussian Elimination: Look deeper into the math behind Gauss-Jordan reduction.
    • Application: Kirchhoff and Circuits: Practice solving linear systems with electrical engineering problems.
  • Vector Spaces: Explore the power of vectors with magic squares, spanning sets, and more...
    • Vector Spaces: What is the essence of a vector?
    • Subspaces and Span: Discover what it means to have a space within a space.
    • Linear Independence: Learn how to spot redundant vectors.
    • Basis and Dimension: What measures the size of a vector space?
    • Dot Products and Inner Products: What's the connection between vectors and geometry?
    • Least Squares: Apply your linear algebra knowledge to an important problem in statistics.
  • Properties of Matrices: A fundamental building block for linear algebra.
    • Matrix Algebra: Find out what matrices and vectors share in common.
    • Inverses and Systems of Equations: Is it ever OK to divide by a matrix?
    • Four Fundamental Subspaces: See what the column, row, null and transpose kernel spaces have to say about a matrix.
    • Adjacency Matrices: Practice fundamental matrix concepts on a graph theory application.
  • Linear Maps and Matrices: Determinants, maps, bases, and more.
    • Linear Transformations: Learn how to turn vectors into... other vectors.
    • Properties of Linear Transformations: Uncover the deep connection between linear transformations and matrices.
    • 2x2 Determinants: Discover a simple test for matrix invertibility.
    • Determinants in Higher Dimensions: Explore some beautiful and useful properties of determinants.
    • Representation by a Matrix: Formalize the relationship between matrices and linear transformations.
    • Change of Basis: What makes two matrices similar?
    • Polynomial Interpolation: Use determinants to fit a polynomial to a collection of data.
  • Eigenvalues and Diagonalizability: Eigenvalues, eigenvectors, and applications!
    • Eigenvalues and Eigenvectors: Learn to work with special vectors of fundamental importance.
    • Characteristic Polynomial: Learn a surefire way to find eigenvalues.
    • Diagonalizability: When can a matrix be put into a diagonal form?
    • PageRank and Exponentiation: Sample some of the uses of diagonalization in graph theory and probability.

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