Linear Algebra for Data, AI & Engineering A Complete Course

Visualize, Compute, and Understand Linear Algebra Like Never Before
What you'll learn
Linear Systems & Solutions: Understand how to model and solve real-world problems using linear equations.
Vector Spaces & Transformations: Explore the building blocks of higher-dimensional spaces and their applications.
Matrices & Operations: Master matrix algebra, inverses, determinants, and factorizations like LU and QR.
Eigenvalues & Eigenvectors: Learn the core concepts behind stability analysis, PCA, and diagonalization.
Orthogonality & Projections: Apply these principles to optimization and least-squares problems.
Advanced Topics: Dive into spectral theorems, affine geometry, and transformations in 2D and 3D.
Requirements
Comfort with high‑school algebra is enough.
Description
Learn the language of modern science and technology-beautifully, clearly, and completely.This course takes you on a carefully designed journey through linear algebra: the mathematics that powers data analysis, computer graphics, optimization, engineering, quantum mechanics, and cryptography. You'll build geometric intuition, computational fluency, and theoretical understanding-so you can solve problems confidently and explain why your methods work.Why this course?Intuition-first, proof-ready: We start with geometry and practical examples, then connect each concept to precise definitions and theorems. You'll see linear algebra before you formalize it.Compute like a pro: From echelon forms and Gauss-Jordan elimination to LU and QR factorizations, you'll learn fast, reliable methods that scale.Think in transformations: Visualize how matrices stretch, rotate, shear, and reflect spaces-and use kernels, images, and rank to reason about them.Beyond the basics: We include fields & modular arithmetic (work over finite fields), affine geometry (top‑down vs. bottom‑up perspectives), and a full tour of orthogonality, projections, and least squares-all the way to spectral theorems and orthogonal diagonalization.Beautiful structure: The course reveals how everything fits: column space, row space, null space, and left null space; basis and dimension; coordinates and change of basis; similarity and matrix representations; determinants and their properties; and the deep equivalences in the Nonsingular Matrix Theorem.What you'll masterLinear Systems: Model real problems and solve them with echelon forms, Gauss-Jordan, and matrix methods-over the reals and over finite fields with modular arithmetic.Vector Spaces & Subspaces: Work fluently with spans, linear combinations, dependence/independence, and classic spaces (columns, rows, functions, matrices).Linear Transformations: Prove linearity, compute kernel and image, and determine when a map is one-to-one or onto. Represent transformations as matrices and change basis cleanly.Matrix Operations & Properties: Multiply, transpose, trace, and reason about edge cases and pitfalls in matrix multiplication.Echelon Forms to Factorizations: Understand elementary matrices, matrix inverses, LU factorization (and its generalizations), and the geometry of matrix transformations.Orthogonality & Projections: Use inner products, norms, distances, and orthogonal complements. Apply Gram-Schmidt, compute orthogonal projections, and solve the least squares problem with geometric clarity.Determinants & Geometry: Compute determinants (cofactors, Laplace expansion), understand their properties and multiplicativity, and connect them to volume, row operations, and Cramer's Rule.Cross Product & Triple Product: Work with 3D geometry, normal vectors, and affine sets.Eigenvalues & Diagonalization: Identify eigenvectors/eigenvalues, compute characteristic polynomials, determine diagonalizability, and perform orthogonal diagonalization for symmetric matrices. Internalize the Spectral Theorem and its implications.Advanced Perspectives: Explore affine transformations, unitary diagonalization, and the connections between similarity, structure, and computation.A uniquely balanced approachTop‑down & bottom‑up: From geometric insight to algebraic structure-and back again-so concepts stick.Worked examples at every step: Solve "Is this vector a linear combination?" in both reals and modular arithmetic; build bases for column and null spaces; compute kernels/images; and check linear independence quickly with principled shortcuts.Concept checks you can feel: Translate theory into skill: manipulate augmented matrices, detect singularity fast, and reason through the Fundamental Spaces of a Matrix without memorizing.By the end, you'll be able to formulate and solve linear systems elegantly, think in vector spaces, convert between bases, factor and analyze matrices, project onto subspaces, solve least squares problems, and diagonalize matrices-with the intuition to see why each move works and the confidence to apply it anywhere.
Who this course is for
Students in mathematics, engineering, computer science, physics, or related fields.
Data/AI professionals seeking a durable, rigorous understanding (PCA via eigen‑analysis, regression via least squares, stability via eigenvalues).
Curious learners who want to truly understand linear algebra-not just "do the steps."
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