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MathMathematical foundations

Linear algebra

Matrices as maps: rank, projections, eigenvectors and matrix factorizations.

3 lessons open · 15 planned

Pages you can open

A matrix moves the plane What does a 2×2 matrix do to the plane? ~30 min Matrix multiplication is composition Which single matrix represents two transformations of the plane applied in sequence? ~30 min What the determinant measures What does the image of the unit square tell you about a 2×2 matrix? ~35 min

Planned

In learning order. These are not pages yet.

  1. Column space, null space, and rankIntro
  2. Dot product and orthogonal projectionIntro
  3. Eigenvectors: the directions a matrix only stretchesIntro
  4. Solving Ax = b: one, none, or infinitely many solutionsIntro
  5. Projection onto a subspace and least squaresIntro
  6. Diagonalization and what happens when you apply A again and againIntro
  7. Why symmetric matrices have an orthogonal basis of eigenvectorsIntro
  8. Gram-Schmidt and the QR decompositionIntro
  9. Quadratic forms: bowls, saddles, and level ellipsesIntro
  10. Singular value decomposition: rotate, stretch, rotateIntro
  11. Gaussian elimination is LU factorizationIntro
  12. What a covariance matrix does to a circleAdvanced undergrad
  13. The matrix exponential e^(At)Advanced undergrad
  14. Positive semidefinite matrices: why a correlation matrix can be invalidAdvanced undergrad
  15. 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.

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