Linear algebra
Matrices as maps: rank, projections, eigenvectors and matrix factorizations.
5 lessons open · 13 planned
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A matrix moves the plane
What does a 2×2 matrix do to the plane?
Matrix multiplication is composition
Which single matrix represents two transformations of the plane applied in sequence?
What the determinant measures
What does the image of the unit square tell you about a 2×2 matrix?
Eigenvectors: directions a matrix only stretches
Which vectors does a matrix keep on their own line through 0?
Diagonalization and powers of a matrix
How do you work out \(A^{20}\) without multiplying \(A\) by itself twenty times?
Planned
In learning order. These are not pages yet.
- Column space, null space, and rankIntro
- Dot product and orthogonal projectionIntro
- Solving Ax = b: one, none, or infinitely many solutionsIntro
- Projection onto a subspace and least squaresIntro
- Why symmetric matrices have an orthogonal basis of eigenvectorsIntro
- Gram-Schmidt and the QR decompositionIntro
- Quadratic forms: bowls, saddles, and level ellipsesIntro
- Singular value decomposition: rotate, stretch, rotateIntro
- Gaussian elimination is LU factorizationIntro
- What a covariance matrix does to a circleAdvanced undergrad
- The matrix exponential e^(At)Advanced undergrad
- Positive semidefinite matrices: why a correlation matrix can be invalidAdvanced undergrad
- Cholesky decomposition: turning independent noise into correlated noiseAdvanced undergrad
How it connects
Builds on Probability Differential equations
Leads to Geometry Stochastic processes Optimization Numerical methods Probability Computational statistics Bonds and interest rates Arbitrage and risk-neutral pricing Calculus Differential equations Regression Equities and portfolios
Subjects marked “connection” are linked by a related idea, not by a prerequisite.