Videos and presentations from lectures LA 2

In comparison with videos, presentations may have some inaccuracies fixed and may include further information (examples, images).

The audio is sometimes not synced well, sorry. I plan to fix it with the next retake.

  1. Determinants
    Determinants, Laplace expansion, adjoint matrix, Cramer's rule (54:10, 464M) - slides only, smaller version (78M)
    The number of spanning trees of a graph (23:50, 65M) - slides only, smaller version (27M)
  2. Polynomials
    Polynomials, Vandermonde matrix, Lagrange interpolation (28:50, 502M) - slides only, smaller version (62M)
  3. Eigenvalues and eigenvectors
    Eigenvalues and eigenvectors of linear maps and matrices, characteristic polynomial (32:52, 887M) - slides only, smaller version (97M)
    Homogeneous systems of first order linear differential equations with constant coefficients (4:35, 8M) - slides only
    Cayley-Hamilton theorem (8:09, 14M) - slides only (updated with an example)
  4. Diagonalization
    Similar matrices, diagonalization, Jordan normal form (21:48, 408M) - slides only (added proof of the Jordan normal form theorem), smaller version (54M)
    Special complex matrices, diagonalization of a Hermitian matrix (19:25, 287M) - slides only, smaller version (33M) May 9, 2022: Fixed the example of orthogonal matrix on page 1.
  5. Inner spaces
    Inner product, norm, Cauchy-Schwarz inequality (20:39, 421M) - slides only, smaller version (25M) (updated, fixed terminology GM -> RMS)
    Orthogonality, orthonormal bases, Fourier coefficients (26:11, 777M) - slides only, smaller version (72M)
    Orthogonal projection, Gram-Schmidt orthonormalization, isometry, orthogonal complement (43:53, 719M) - slides only, smaller version (93M)
  6. Positive definite matrices
    Positive definite matrices, Cholesky factorization, Sylvester's condition (33:58, 634M) - slides only, smaller version (78M)
  7. Bilinear and quadratic forms
    Bilinear and quadratic forms, their matrices, Sylvester's law of inertia (43:38, 672M) - slides only, smaller version (93M)
    May 3, 2022: updated presentation - proof of the theorem about diagonalization of q. forms is rewritten so that the transposed matrix is on the left
    Conics and quadrics (6:19, 11M) - slides only, smaller version (5M)
  8. Applications
    The maximum number of lines spanning the same angle (9:07, 191M) - slides only, smaller version (17M)

See also videos from the winter term.

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