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.

Overview

Section 1: Linear Systems

Lecture 1 Linear Systems

Lecture 2 Number of Solutions to a Linear System

Lecture 3 Three Dimensional Linear Systems and Homogeneous Systems

Section 2: Fields

Lecture 4 What is a Number? A Precursor to the Notation of an Algebraic Field.

Lecture 5 Solving a Linear Equation over a Field

Lecture 6 Modular Arithmetic

Lecture 7 Solving a Linear Equation over a Finite Field

Section 3: Vector Spaces

Lecture 8 Vector Spaces

Lecture 9 The Vector Space F^n and Column Vectors

Lecture 10 Linear Combinations

Lecture 11 The Vector Space of Linear Equations

Lecture 12 Properties of a Vector Spaces

Section 4: Linear Transformations

Lecture 13 Linear Transformations

Lecture 14 Proving a Transformation is Linear

Lecture 15 Computing the Kernel of a Linear Transformation

Lecture 16 Computing the Image of a Linear Transformation

Lecture 17 One-to-One and Onto Linear Transformations

Section 5: Augmented Matrices

Lecture 18 Elementary Row Operations

Lecture 19 Augmented Matrices

Lecture 20 Echelon Forms

Section 6: Reduction of Linear Systems

Lecture 21 Solving Linear Systems using Echelon Forms

Lecture 22 Gauss-Jordan Elimination

Section 7: Vector Equations

Lecture 23 Vector Equations

Lecture 24 Is a Vector a Linear Combination?

Lecture 25 Is a Vector a Linear Combination? (Modular Arithmetic)

Lecture 26 The Span of Vectors

Section 8: Matrix Equations

Lecture 27 Matrix-Vector Multiplication

Lecture 28 Matrix Equations

Lecture 29 The Column Space

Lecture 30 Matrix Transformations

Section 9: Linear Independence

Lecture 31 Linear Independence

Lecture 32 Determining Linear Dependence

Lecture 33 Checking Linear Independence Quickly!

Section 10: Affine Geometry

Lecture 34 Affine Geometry

Lecture 35 The Top-Down Approach

Lecture 36 The Bottom-Up Approach

Lecture 37 Equations of Affine Sets

Lecture 38 Affine Combinations

Lecture 39 Intersections of Affine Sets

Section 11: Subspaces

Lecture 40 Subspaces

Lecture 41 Not Every Subset is a Subspace

Lecture 42 The Column Space is a Subspace

Lecture 43 Function Spaces

Section 12: Solution Sets of Linear Systems

Lecture 44 Solutions Sets of Homogeneous Linear Systems

Lecture 45 The Null Space

Lecture 46 Solutions Sets of Non-Homogeneous Linear Systems

Section 13: Bases

Lecture 47 Basis and Dimension of a Vector Space

Lecture 48 Finding a Basis of the Column Space

Lecture 49 Finding a Basis for the Null Space

Lecture 50 Find a Basis for the Column Space and Null Space QUICKLY

Section 14: Coordinates

Lecture 51 Coordinate Vectors

Lecture 52 The Change-of-Basis Matrix

Lecture 53 Computing the Change-of-Basis Matrix

Section 15: Matrix Operations

Lecture 54 The Vector Space of Matrices

Lecture 55 Matrix Multiplication

Lecture 56 The Transpose of a Matrix

Lecture 57 The Trace of a Matrix

Section 16: Matrix Properties

Lecture 58 Properties of Matrix Operations

Lecture 59 Problems with Matrix Multiplication

Lecture 60 The Row Space

Section 17: Matrix Inverses

Lecture 61 Matrix Inverses

Lecture 62 2 x 2 Matrix Inverses

Lecture 63 Properties of Matrix Inversion

Lecture 64 The Nonsingular Matrix Theorem

Section 18: Elementary Matrices

Lecture 65 Elementary Matrices

Lecture 66 Elementary Factorizations of Matrices

Section 19: Matrix Factorizations

Lecture 67 Diagonal Matrices

Lecture 68 Permutation Matrices

Lecture 69 Triangular Matrices

Lecture 70 LU Factorization

Lecture 71 Solving a Linear System using the LU Factorization

Lecture 72 Generalizations of the LU Factorization

Section 20: Linear Transformations on R^2

Lecture 73 Dilations and Contractions in the Plane (Linear Algebra)

Lecture 74 Shearing in the Plane (Linear Algebra)

Lecture 75 Reflections in the Plane (Linear Algebra)

Lecture 76 Rotations in the Plane (Linear Algebra)

Lecture 77 Determining the Geometric Transformations of a 2 x 2 Matrix

Section 21: Representations of Linear Transformations as Matrices

Lecture 78 Finding the Kernel of a Linear Transformation

Lecture 79 Find the Image of a Linear Transformation

Lecture 80 Matrix Representations of Linear Transformation

Section 22: Inner Products

Lecture 81 Inner Products

Lecture 82 Norms of Vectors

Lecture 83 Distance Between Vectors

Section 23: Orthogonality

Lecture 84 Orthogonal Vectors

Lecture 85 Normal Vectors

Lecture 86 Orthogonal Sets

Lecture 87 Orthogonal Complements

Section 24: Outer Products

Lecture 88 Symmetric and Hermitian Matrices

Lecture 89 Projections and Idempotent Matrices

Lecture 90 Outer Products

Lecture 91 Nilpotent Matrices

Section 25: Affine Transformations

Lecture 92 Angles Between Vectors

Lecture 93 Orthogonal Matrices

Lecture 94 Affine Transformations

Section 26: Orthogonal Projections

Lecture 95 Fourier Coefficients of an Orthogonal Basis

Lecture 96 Orthogonal Projections

Lecture 97 The Orthogonal Decomposition Theorem

Lecture 98 The Best Approximation Theorem

Section 27: The Fundamental Theorem of Linear Algebra

Lecture 99 The Fundamental Spaces of a Matrix

Lecture 100 The Left Null Space

Lecture 101 The Fundamental Theorem of Linear Algebra

Section 28: The Gram-Schmidt Algorithm

Lecture 102 Orthogonalizing a Complex Basis

Lecture 103 QR Factorization

Section 29: The Least Squares Problem

Lecture 104 The Least Squares Problem

Lecture 105 The General Solution of the Least Squares Problem

Section 30: Determinants

Lecture 106 Determinants

Lecture 107 Cofactors and Laplace Expansion

Section 31: Properties of Determinants

Lecture 108 Multilinear Transformations

Lecture 109 The Multiplicative Principle of Determinants

Lecture 110 Determinants and Row Reduction

Section 32: Cramer's Rule

Lecture 111 Cramer's Rule

Lecture 112 Solving a 2 x 2 System with Cramer's Rule

Lecture 113 The Adjugate of a Matrix

Section 33: The Cross Product

Lecture 114 The Cross Product

Lecture 115 The Scalar Triple Product

Lecture 116 Properties of the Cross Product

Lecture 117 Normal Vectors and Affine Sets

Section 34: Eigenvalues and Eigenvectors

Lecture 118 What is an Eigenvalue?

Lecture 119 How to check if a Vector is an Eigenvector?

Lecture 120 How to check if a Scalar is an Eigenvalue?

Lecture 121 Finding a Basis for the Eigenspace of a Matrix

Lecture 122 Finding the Eigenvalues of a Triangular Matrix

Section 35: The Characteristic Polynomial

Lecture 123 What is a Characteristic Polynomial of a Matrix?

Lecture 124 Computing the Characteristic Polynomial of a Matrix

Lecture 125 Computing the Characteristic Polynomial of a Matrix with non-real Eigenvalues

Lecture 126 Nonsingular Matrices and Eigenvalues

Lecture 127 Similar Matrices

Section 36: Diagonalization

Lecture 128 Linear Independence of Eigenvectors

Lecture 129 Diagonalizable Matrices

Lecture 130 Matrix Diagonalization - The Whole Enchilada!

Section 37: Orthogonal Diagonalization

Lecture 131 Eigenvectors and Symmetric Matrices

Lecture 132 A Diagonalization of a Symmetric Matrix

Lecture 133 Orthogonally Diagonalizable Matrices

Lecture 134 Orthogonal Diagonalization of a Symmetric Matrix

Lecture 135 The Spectral Theorem of Symmetric Matrices

Lecture 136 Unitary Diagonalization - The Whole Enchilada!

Lecture 137 Spectral Decomposition of a Symmetric Matrix

Section 38: Similarity and Linear Transformations

Lecture 138 Matrix Representations of a Linear Transformation for any Basis

Lecture 139 Linear Transformations and Similarity

Section 39: Additional Topics

Lecture 140 Equivalence of Unitary Matrices

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.”