Linear Algebra
Algebra
Matrices, vector spaces and linear maps, determinants, eigenvalues and diagonalization, orthogonality and least squares, quadratic forms, the SVD, and Markov chains.
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Matrices
Unit 1 · Matrices & Systems
0 of 11 units complete
- 1Unit 1Matrices & Systems5 parts · about 11 min
Matrices
A matrix is a machine that moves the plane. Multiply matrices (order matters!), see a system of equations two ways — crossing lines and combining columns — undo a matrix with its inverse, and get Fibonacci numbers by squaring.
- 2Unit 2Row Reduction & Rank5 parts · about 10 min
Row Reduction
Solve any system of linear equations with three moves: swap, scale, and add a multiple. Reduce a 3 × 3 system by hand, meet free variables and rank, and find out how Gauss's method secretly factors a matrix.
- 3Unit 3Determinants & Inverses5 parts · about 12 min
Determinants
One number that tells you how much a matrix stretches area, whether it flips the plane over, and whether it can be undone. Squash a square flat, track what row operations do, solve systems with Cramer's rule, and find a triangle's area in one line.
- 4Unit 4Vector Spaces & Bases5 parts · about 12 min
Vector Spaces
Everything you can build from a few vectors by stretching and adding. Reach any point of the plane with two arrows (unless they line up), test which sets are subspaces, and measure the same point in a new coordinate system.
- 5Unit 5Linear Transformations5 parts · about 11 min
Linear Transformations
Stretches, shears, reflections, rotations and projections are all matrices. Build each one by deciding where two arrows land, test which maps are linear, and see why composing maps means multiplying matrices.
- 6Unit 6Orthogonality & Least Squares5 parts · about 11 min
Orthogonality
Right angles make everything easy. Test perpendicularity with the dot product, drop the shortest line onto a vector, straighten any basis with Gram–Schmidt, and fit the best line through data that no line fits exactly.
- 7Unit 7Eigenvalues5 parts · about 13 min
Eigenvalues
Most vectors get knocked off their line by a matrix. A few special ones only stretch. Hunt for them by dragging, compute them from a characteristic polynomial, and find out why a rotation has none (in the real plane).
- 8Unit 8Diagonalization & Matrix Powers5 parts · about 10 min
Diagonalization
In the right coordinates every (nice) matrix is just a stretch along axes. Use that to take the 100th power of a matrix in your head, watch powers line up with the dominant eigenvector, and get a formula for Fibonacci numbers.
- 9Unit 9Symmetric Matrices & Quadratic Forms5 parts · about 11 min
Symmetric Matrices
Symmetric matrices are the best-behaved matrices there are: real eigenvalues, perpendicular eigenvectors. Use them to draw ellipses and saddles, rotate a tilted ellipse straight, and find the biggest value of a quadratic on a circle.
- 10Unit 10Singular Values & the SVD5 parts · about 11 min
The SVD
Every matrix, symmetric or not, square or not, is a rotation, a stretch along axes, and another rotation. See the unit circle become an ellipse, read the singular values off its axes, and compress an image by throwing most of a matrix away.
- 11Unit 11Markov Chains & Long-Run Behavior5 parts · about 12 min
Markov Chains
Random processes that only remember where they are now. Forecast the weather many days ahead, find the steady state as an eigenvector, rank web pages like early Google, and work out where a gambler ends up.