Advanced Linear Algebra

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• What a field is ~18 min
• Vector spaces over ℚ, ℝ, ℂ, and 𝔽ₚ ~18 min
• Familiar spaces reread in field-general language ~16 min
• Infinite-dimensional spaces: polynomials, sequences, functions ~16 min
• The exchange lemma and why dimension is well-defined ~18 min
• Sums of subspaces and the dimension formula ~18 min
• Direct sums ~16 min
• Complements and the projections they define ~16 min
• Quotient spaces V/W ~18 min
• The first isomorphism theorem for vector spaces ~18 min

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• L(V,W) is itself a vector space ~16 min
• Rank-nullity, proved from the quotient ~14 min
• Direct sums of maps and block matrices ~16 min
• Invariant subspaces ~14 min
• Block-triangular form from an invariant subspace ~16 min
• The algebra L(V): composition, powers, polynomials in T ~16 min
• Nilpotent operators, first look ~14 min
• Idempotents and projections (E² = E) ~16 min
• What similarity actually classifies ~16 min

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• Linear functionals ~14 min
• The dual space V* ~14 min
• Dual bases ~16 min
• Coordinates are functionals ~14 min
• The dual of an infinite-dimensional space ~14 min
• Annihilators ~14 min
• Annihilators and dimension ~14 min
• The dual (transpose) map T* ~16 min
• The matrix of the dual map is the transpose ~16 min
• The double dual and natural vs. non-natural isomorphism ~18 min

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• Bilinear forms ~15 min
• The Gram matrix of a form ~15 min
• Change of basis for forms: congruence, not similarity ~16 min
• Symmetric and alternating forms ~15 min
• Quadratic forms and polarization ~16 min
• Diagonalizing a form by completing the square ~17 min
• Sylvester's law of inertia ~16 min
• Signature and the classification of real quadratic forms ~15 min
• Nondegenerate forms and orthogonal complements ~16 min
• Sesquilinear and Hermitian forms over C ~17 min

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• Inner products over R and C ~16 min
• Norms from inner products, and norms that aren't ~15 min
• Cauchy-Schwarz and the triangle inequality, proved ~16 min
• Orthonormal bases and Fourier coefficients ~16 min
• Gram-Schmidt in a function space: Legendre polynomials ~17 min
• Orthogonal complements and V = W (+) W-perp ~16 min
• Orthogonal projection and best approximation ~16 min
• Least squares as best approximation ~15 min
• The Riesz representation theorem ~17 min
• The adjoint operator T* ~17 min
• The four fundamental subspaces via the adjoint ~16 min

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• Every operator on a complex space has an eigenvalue ~16 min
• Invariant subspaces and upper-triangular form ~16 min
• Schur's theorem ~17 min
• The minimal polynomial ~16 min
• Minimal versus characteristic polynomial ~16 min
• Diagonalizability and the minimal polynomial's roots ~16 min
• Self-adjoint operators and real eigenvalues ~15 min
• The real spectral theorem ~17 min
• Normal operators ~15 min
• The complex spectral theorem ~17 min
• Unitary and orthogonal operators ~16 min
• Spectral decomposition as a sum of projections, and simultaneous diagonalization ~17 min

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• Why diagonalization can fail ~15 min
• Generalized eigenvectors ~15 min
• Generalized eigenspaces ~15 min
• Nilpotent operators and their structure ~16 min
• Jordan chains ~16 min
• The Jordan canonical form ~17 min
• Computing the Jordan form from rank counts and dot diagrams ~17 min
• The Jordan-Chevalley (S + N) decomposition ~16 min
• Primary decomposition ~17 min
• Cyclic subspaces and companion matrices ~16 min
• Rational canonical form, invariant factors, and elementary divisors ~17 min

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• Polynomials of an operator, revisited ~15 min
• The Cayley-Hamilton theorem ~16 min
• Cayley-Hamilton via the Jordan form ~15 min
• The spectral mapping theorem ~15 min
• Inverses from polynomials ~15 min
• Functions of a diagonalizable matrix ~16 min
• Functions of a general matrix, via Jordan blocks ~17 min
• The matrix exponential ~17 min
• Matrix logarithms and square roots ~16 min
• Linear systems x' = Ax, stability, and phase portraits ~17 min

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• What each factorization actually buys ~15 min
• LU and Cholesky as a positive-definiteness certificate ~16 min
• QR revisited: Householder and Givens versus Gram-Schmidt ~16 min
• Schur decomposition, computed ~15 min
• Normal matrices and unitary diagonalization as a Schur corollary ~15 min
• Rank factorization A = CR ~15 min
• The SVD: statement and existence ~17 min
• Polar decomposition ~15 min
• The pseudoinverse ~16 min
• The Moore-Penrose conditions and least squares, again ~16 min

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• Singular values and singular vectors ~16 min
• SVD geometry: rotate, stretch, rotate ~16 min
• SVD versus eigendecomposition ~16 min
• The four fundamental subspaces read off the SVD ~16 min
• Rank, numerical rank, and the condition number ~16 min
• The Eckart-Young theorem ~17 min
• Low-rank approximation in practice ~16 min
• Image compression ~15 min
• PCA is an SVD ~17 min
• Latent structure: recommenders, embeddings, and signals ~15 min

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• Quadratic forms as surfaces ~15 min
• Conics, quadrics, and the principal-axis theorem ~16 min
• Ellipsoids and the shape of a positive definite matrix ~15 min
• Positive definite and semidefinite matrices ~15 min
• Four tests for definiteness, and when each is the right one ~16 min
• The Rayleigh quotient ~15 min
• Rayleigh-Ritz and the extreme eigenvalues ~16 min
• The Courant-Fischer min-max theorem ~17 min
• Eigenvalue interlacing and Weyl's inequality ~17 min
• Constrained quadratic optimization and the generalized eigenvalue problem ~17 min

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• Operators on function spaces ~15 min
• Matrix norms and operator norms ~16 min
• The induced 2-norm and the spectral radius ~16 min
• Gelfand's formula and when A^n converges to zero ~16 min
• The Neumann series and the resolvent ~16 min
• Conditioning and the perturbation of eigenvalues ~17 min
• Approximate eigenvectors and why the spectrum can be unstable ~17 min
• Orthogonal versus oblique projections ~16 min
• Compact operators: an introductory look ~16 min
• What actually breaks in infinite dimensions ~17 min

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• Multilinear maps ~15 min
• The tensor product and its universal property ~17 min
• Bases and coordinates for tensors ~16 min
• Covariant, contravariant, and mixed tensors ~17 min
• Contraction and the trace ~16 min
• Alternating multilinear forms ~15 min
• The wedge product ~16 min
• Exterior powers Λᵏ V and the dimension count ~16 min
• Determinants as alternating multilinear forms ~17 min
• Oriented area, volume, and orientation ~15 min
• The Grassmann algebra and where differential forms pick up ~15 min