Linear Algebra
Strang's course, animated: matrices that move space, eigenvectors that hold their line, and least squares as a shadow on a wall.
12 units · 120 topics · 562 questions
Vectors as arrows, the dot product as an angle detector, and span as the whole territory a handful of arrows can reach.
What a Vector Is · Adding and Scaling Vectors · Linear Combinations of Vectors · The Standard Basis Vectors · +6 more
Two ways to picture a system of equations, and Gaussian elimination as the mechanical process that solves either picture.
The Row Picture of a Linear System · The Column Picture of a Linear System · Gaussian Elimination, Step by Step · Elimination Using Matrices · +8 more
How matrices multiply, when they can be undone, and how elimination itself gets packaged into the factorization A = LU.
Matrix-Vector Multiplication, Two Ways · Matrix-Matrix Multiplication · Matrix Multiplication Is Not Commutative · The Transpose of a Matrix · +10 more
Every matrix carries four hidden spaces around with it — the column space, null space, row space, and left null space.
What a Vector Space Is · Subspaces of a Vector Space · The Column Space of a Matrix · The Null Space of a Matrix · +11 more
Dropping a perpendicular onto a subspace, and why that single idea is the best-fit answer to a system with no exact solution.
Orthogonal Vectors · Orthogonal Complements of a Subspace · Orthogonality of the Four Subspaces · Orthonormal Bases · +8 more
One number that measures how much a matrix scales area or volume, flips orientation, or collapses space entirely.
What a Determinant Measures · The 2x2 and 3x3 Determinant Formulas · Properties of Determinants · Cofactor Expansion · +4 more
The vectors a matrix refuses to knock off their own line, and how they predict everything a system does over time.
What an Eigenvector Is · The Characteristic Polynomial · Finding Eigenvalues and Eigenvectors · The Trace and Determinant Shortcut for Eigenvalues · +8 more
Why symmetric matrices always diagonalize using perpendicular directions alone, and what makes a quadratic bowl curve upward.
Symmetric Matrices and Real Eigenvalues · The Spectral Theorem · Orthogonal Diagonalization · Positive Definite Matrices · +2 more
A matrix as a machine that moves the whole plane at once, and how the same arrow looks different described in two different grids.
What a Linear Transformation Is · The Matrix of a Linear Transformation · Rotation, Reflection, and Scaling as Matrices · Shear Transformations · +5 more
Any matrix at all, even a non-square one, unpacked into a rotation, a stretch along perpendicular axes, and another rotation.
The Idea of the SVD · Computing a Singular Value Decomposition · Singular Values and the Unit Circle Becoming an Ellipse · The Pseudoinverse · +3 more
What changes when a vector's entries turn complex, and how a matrix whose columns sum to 1 predicts where a population settles.
Complex Vectors and the Hermitian Dot Product · Complex Numbers as 2x2 Rotation Matrices · Complex Eigenvalues of Real Matrices · Markov Matrices and Transition Probabilities · +2 more
Where all of it lands: circuits and networks as linear systems, data compressed by its dominant directions, and shapes moved by matrices.
Graphs and Incidence Matrices · The Null Space of an Incidence Matrix · Kirchhoff's Laws as a Linear System · Computer Graphics and 2D Transformations · +5 more