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# Linear Algebra with Applications

via Brilliant

### 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.

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