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Linear Algebra

Vectors, transformations, eigenvalues, and orthogonality.

Prerequisites: Calculus lessons 1–3

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  1. 1Vectors and Dot ProductsVectors as arrows and coordinates, the dot product, orthogonality, and a classic correlation puzzle.
  2. 2Matrices as TransformationsMatrix-vector multiplication as a linear map: rotations, scalings, projections, and composition.
  3. 3Systems of Linear EquationsAx = b, Gaussian elimination, and when solutions exist or are unique.
  4. 4Independence, Basis, and DimensionLinear independence, span, and why every basis of a space has the same size.
  5. 5Rank, Nullity, and the Four SubspacesColumn space, null space, the rank-nullity theorem, and the fundamental picture of a matrix.
  6. 6DeterminantsThe determinant as a volume-scaling factor, its properties, and the invertibility test.
  7. 7Eigenvalues and EigenvectorsThe directions a matrix preserves, the characteristic polynomial, and trace/det identities.
  8. 8OrthogonalityProjections, orthonormal bases, orthogonal matrices, and least squares.
  9. 9Diagonalization and the Spectral TheoremWrite A = PΛP⁻¹ to take any power of a matrix in one step, use the orthogonal version A = QΛQᵀ for symmetric matrices, and find PageRank as the eigenvector with eigenvalue 1.
  10. 10SVD, QR, Cholesky and Positive DefinitenessPositive definite matrices and why covariance matrices are PSD, Cholesky for simulating correlated normals, QR for stable least squares, and the SVD as rotate, stretch, rotate.
  11. 11Principal Component AnalysisPrincipal components as eigenvectors of the covariance matrix, computing them with the SVD, choosing how many to keep, and the level, slope and curvature of the yield curve.

Practice

The problem bank has 21 interview problems on this material, with hints and full solutions.

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