Linear Algebra with Applications

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Linear Algebra with Applications provided by Brilliant is a comprehensive online course, which lasts for 30-40 hours worth of material. Upon completion of the course, you can receive an e-certificate from Brilliant. The course is taught in Englishand is Paid Course. Visit the course page at Brilliant for detailed price information.

Overview
  • 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.
    The course also includes applications quizzes with topics drawn from such diverse areas as image compression, cryptography, error coding, chaos theory, and probability.

Syllabus
    • Introduction to Vector Spaces:
      • What is a Vector?: Discover the true nature of vectors.
      • Waves as Abstract Vectors: Take a visual tour of vector spaces.
      • Why Vector Spaces?: Experience the power of abstraction.
    • System of Equations:
      • The Gauss-Jordan Process I: Gain experience with systems of equations through traffic planning.
      • The Gauss-Jordan Process II: Learn a surefire method for cracking any set of linear equations.
      • Application: Markov Chains I: Apply your Gauss-Jordan skills to a classic probability problem.
    • Vector Spaces:
      • Real Euclidean Space I: Learn about important abstract concepts in a familiar setting.
      • Real Euclidean Space II: Lay the foundation for building vector spaces.
      • Span & Subspaces: Develop a quick means for generating vector spaces.
      • Coordinates & Bases: Condense common vector spaces with bases.
      • Matrix Subspaces: Uncover the deep connection between the null and column spaces of a matrix.
      • Application: Coding Theory: Discover how linear algebra is used in error-correction schemes.
      • Application: Graph Theory I: Unravel the properties of graphs with linear algebra.
    • Linear Transformations:
      • What Is a Matrix?: Free your mind from viewing matrices as just arrays of numbers.
      • Linear Transformations: Come full circle and connect linear maps back to matrices.
      • Matrix Products: Find out one way of multiplying matrices together.
      • Matrix Inverses: Learn when it's OK to divide by a matrix.
      • Application: Image Compression I: Use linear algebra to store and transmit pictures efficiently.
      • Application: Cryptography: Crack secret messages with linear algebra!
    • Multilinear Maps & Determinants:
      • Bivectors: Take the first step towards determinants with bivectors.
      • Trivectors & Determinants: Evaluate determinants like a pro with trivectors.
      • Determinant Properties: Gain experience with the most important properties of determinants.
      • Multivector Geometry: Learn about the visual aspects of multivectors.
      • Dual Space: Create new vector spaces from old ones using the "dual" concept.
      • Tensors & Forms: Acquaint yourself with tensors, a cornerstone of modern geometry.
      • Tensor Products: Open up new frontiers with tensor multiplication.
      • Wedges & Determinants: Practice calculating determinants with wedge products.
    • Eigenvalues & Eigenvectors:
      • Application: Markov Chains II: Discover eigenvectors by rethinking a classic probability problem.
      • Eigenvalues & Eigenvectors: Learn the essentials of eigenvalues & eigenvectors.
      • Diagonalizability: Restructure square matrices in the most useful way imaginable.
      • Normal Matrices: When can a matrix be diagonalized?
      • Jordan Normal Form: Explore the next best thing to diagonalization.
      • Application: Graph Theory II: Use your eigen-knowledge to uncover deep properties of graphs.
      • Application: Discrete Cat Map: Connect chaos with linear algebra.
      • Application: Arnold's Cat Map: See how eigenvalues & eigenvectors quantify unpredictability.
    • Inner Product Spaces:
      • Inner Product Spaces: Extend familiar geometric tools to abstract spaces.
      • Gram-Schmidt Process: Practice making your very own orthonormal bases.
      • Least Squares Regression: Solve a crucial problem in statistics with inner product spaces.
      • Singular Values & Vectors: Build singular values & vectors with least squares regression.
      • Singular Value Decompositions: Find out how to "diagonalize" a non-square matrix.
      • SVD Applications: Compress data with singular value decompositions.