Linear Algebra II

Video 2

field of scalars, complex conjugate, modulus of a complex number, vector space axioms, subspace test, real and complex dimension and polynomials of degree at most n

Builds on Proof Strategies in Linear Algebra. Also uses vector addition, scalar multiplication and basic algebra.

Video 3

exchange lemma, invariance of basis size, dimension, extracting a basis from a spanning list, extending an independent list to a basis and dimension of a subspace

Builds on Proof Strategies in Linear Algebra and Real and Complex Vector Spaces. Also uses span, linear independence and bases from a first linear algebra course.

Video 4

space of linear maps, pointwise addition and scaling of maps, linear extension from basis images, matrix-unit maps, dimension of the space of linear maps and rank-one map

Builds on Basis Exchange and Dimension.

Video 5

rank-nullity theorem, injective map, surjective map, isomorphism, linear inverse and rank of a composition

Builds on The Vector Space of Linear Maps and Basis Exchange and Dimension.

Video 6

sum of subspaces, intersection of subspaces, dimension formula for a sum, union of subspaces and product of vector spaces

Builds on Basis Exchange and Dimension and Rank-Nullity and Isomorphisms.

Video 7

direct sum, uniqueness of decomposition, complementary subspace, projection along a complement, idempotent map and oblique projection

Builds on Sums and Intersections of Subspaces and The Vector Space of Linear Maps.

Video 8

equivalence modulo a subspace, coset, quotient space, well-defined operation, quotient basis and dimension of a quotient space

Builds on Sums and Intersections of Subspaces and Proof Strategies in Linear Algebra.

Video 9

fiber of a linear map, quotient map, induced map, first isomorphism theorem and polynomial evaluation map

Builds on Quotient Spaces, Rank-Nullity and Isomorphisms and Direct Sums and Projections.

Video 10

linear functional, dual space, dual basis, coordinate functional, evaluation functional and conjugate-linear map

Builds on Basis Exchange and Dimension and The Vector Space of Linear Maps.

Video 11

covector, pullback of a functional, dual map, dual map of a composition, matrix of the dual map and covector coordinate change

Builds on Dual Spaces and Dual Bases and Rank-Nullity and Isomorphisms. Also uses change-of-basis matrices from a first linear algebra course.

Video 12

annihilator, dimension of an annihilator, kernel of the dual map, image of the dual map, row space and left nullspace and annihilator versus orthogonal complement

Builds on Dual Spaces and Dual Bases, Dual Maps and Covectors and Rank-Nullity and Isomorphisms. Also uses the four fundamental subspaces of a matrix.

Video 13

multilinear form, alternating form, normalized determinant, sign of a permutation, uniqueness of the determinant and multiplicativity of the determinant

Builds on Basis Exchange and Dimension and The Vector Space of Linear Maps. Also uses determinant calculations from a first linear algebra course.

Video 14

trace, cyclic property of the trace, similar matrices, similarity invariant, characteristic polynomial and real volume factor of a complex matrix

Builds on Determinants and Multilinear Forms, Rank-Nullity and Isomorphisms and Dual Maps and Covectors. Also uses similar matrices and eigenvalues from a first linear algebra course.

Video 15

bilinear form, matrix of a bilinear form, congruent matrices, quadratic form, polarization identity and symmetric and skew-symmetric parts

Builds on Dual Spaces and Dual Bases, Dual Maps and Covectors and Trace, Determinant and Similarity.

Video 16

conjugate transpose, complex inner product, conjugate symmetry, positive definiteness, sesquilinear form, norm from an inner product and Gram matrix

Builds on Real and Complex Vector Spaces and Bilinear Forms and Congruence. Also uses the dot product and length in real coordinate space.

Video 17

Cauchy-Schwarz inequality, equality case of Cauchy-Schwarz, triangle inequality, complex Gram-Schmidt process, orthonormal expansion, Parseval's identity and Bessel's inequality

Builds on Complex Inner Products and Basis Exchange and Dimension. Also uses projection onto a line and the real Gram-Schmidt process.

Video 18

Riesz representation theorem, representing vector of a functional, construction from an orthonormal basis, conjugate-linear identification and dependence on the chosen inner product

Builds on Dual Spaces and Dual Bases and Cauchy-Schwarz and Orthonormal Expansions.

Video 19

adjoint of a linear map, adjoint identity, adjoint of a composition, matrix of the adjoint in a general basis, orthogonal complement, double orthogonal complement and kernel and image of the adjoint

Builds on Riesz Representation in Finite Dimensions, Dual Maps and Covectors and Complex Inner Products.

Video 20

orthogonal decomposition, orthogonal projection, best approximation, projection theorem, orthogonal projector and polynomial approximation in an inner product space

Builds on Direct Sums and Projections, Cauchy-Schwarz and Orthonormal Expansions and Adjoints and Gram Matrices.

Video 21

weighted inner product, positive definite weight matrix, weighted orthogonality, weighted normal equations, uniqueness of the fitted vector and inner product from sample values

Builds on Orthogonal Projection and Best Approximation and Bilinear Forms and Congruence. Also uses least squares and the normal equations from a first linear algebra course.

Video 22

invariant subspace, restriction of an operator, adapted basis, block upper triangular matrix, block diagonal matrix and reducing subspace

Builds on Direct Sums and Projections, Trace, Determinant and Similarity and Orthogonal Projection and Best Approximation.

Video 23

Schur decomposition, unitary matrix, compression of an operator, induction on dimension, algebraic multiplicity, trace as a sum of eigenvalues and determinant as a product of eigenvalues

Builds on Invariant Subspaces and Block Matrices, Cauchy-Schwarz and Orthonormal Expansions, Determinants and Multilinear Forms and Proof Strategies in Linear Algebra. Also uses the fundamental theorem of algebra, taken without proof.

Video 24

complex conjugate eigenpair, real invariant plane, real Schur form, two-by-two real Schur block and rotation-scaling block

Builds on Schur Decomposition and Real and Complex Vector Spaces.

Video 25

normal operator, Hermitian operator, unitary operator, complex spectral theorem, unitary diagonalization, real eigenvalues of a Hermitian operator and eigenvalues of modulus one

Builds on Schur Decomposition and Adjoints and Gram Matrices.

Video 26

commuting operators, simultaneous diagonalization, common orthonormal eigenbasis, eigenspace preserved by a commuting operator, normal restriction to a reducing subspace and commuting matrices that are not normal

Builds on The Spectral Theorem for Normal Operators and Invariant Subspaces and Block Matrices.

Video 27

diagonalization by congruence, conjugate congruence, inertia, maximal positive subspace, Sylvester's law of inertia and indefinite form

Builds on Bilinear Forms and Congruence and The Spectral Theorem for Normal Operators.

Video 28

positive definite matrix, positive semidefinite matrix, completing the square, Cholesky factorization, Schur complement, leading principal minor and Sylvester's criterion

Builds on Quadratic Forms and Sylvester's Law of Inertia and Weighted Least Squares. Also uses Gaussian elimination and pivots.

Video 29

Rayleigh quotient, scale invariance of the quotient, weighted average of eigenvalues, minimum and maximum of the Rayleigh quotient and extremizers in a repeated eigenspace

Builds on The Spectral Theorem for Normal Operators and Quadratic Forms and Sylvester's Law of Inertia.

Video 30

eigenvalues in increasing order, min-max principle, max-min principle, dimension count for intersecting subspaces and attaining subspace

Builds on Rayleigh Quotients and Sums and Intersections of Subspaces.

Video 31

Gram operator, singular value, left and right singular vectors, singular value decomposition, thin and compact SVD and nonuniqueness of singular vectors

Builds on The Spectral Theorem for Normal Operators, Orthogonal Projection and Best Approximation and Quadratic Forms and Sylvester's Law of Inertia. Also uses singular value decomposition calculations from a first linear algebra course.

Video 32

pseudoinverse, least-squares solution set, minimum-norm least-squares solution, projector onto the column space, projector onto the row space and complementary projectors

Builds on The Singular Value Decomposition and Orthogonal Projection and Best Approximation.

Video 33

Moore-Penrose conditions, Hermitian idempotent, uniqueness of the pseudoinverse, pseudoinverse of a singular matrix and generalized inverse

Builds on The Pseudoinverse and Four Projectors and Adjoints and Gram Matrices.

Video 34

spectral norm, Frobenius norm, truncated SVD, Eckart-Young theorem, best low-rank approximation and nonuniqueness for repeated singular values

Builds on The Singular Value Decomposition, The Min-Max Principle and Rank-Nullity and Isomorphisms.

Video 35

centered data matrix, sample covariance matrix, principal components, projected variance, reconstruction error and effect of centering and units

Builds on The Eckart-Young Theorem and Orthogonal Projection and Best Approximation. Also uses the arithmetic mean of a list of numbers.

Video 36

positive semidefinite square root, spectral construction of a square root, uniqueness of the positive root, square roots of a matrix and Cholesky factor versus square root

Builds on The Spectral Theorem for Normal Operators and Quadratic Forms and Sylvester's Law of Inertia.

Video 37

absolute value of a matrix, polar decomposition, partial isometry, isometry, unitary extension and polar form with the positive factor on the left

Builds on Positive Square Roots of Matrices, The Singular Value Decomposition and The Pseudoinverse and Four Projectors.

Video 38

polynomial in an operator, annihilating polynomial, minimal polynomial, polynomial division and roots of the minimal polynomial

Builds on The Vector Space of Linear Maps, Invariant Subspaces and Block Matrices and Trace, Determinant and Similarity. Also uses roots and factors of polynomials in one variable.

Video 39

adjugate identity, Cayley-Hamilton theorem, polynomial with matrix coefficients, reduction of high powers and characteristic versus minimal polynomial

Builds on Matrix Polynomials and the Minimal Polynomial, Determinants and Multilinear Forms and Schur Decomposition. Also uses cofactors and the adjugate matrix.

Video 40

polynomial Euclidean algorithm, Bézout's identity, coprime polynomials, polynomial projector, distinct-root criterion for diagonalizability and simultaneous diagonalization of commuting diagonalizable operators

Builds on Matrix Polynomials and the Minimal Polynomial, The Cayley-Hamilton Theorem and Rank-Nullity and Isomorphisms.

Video 41

orbit of a vector, cyclic subspace, cyclic vector, companion matrix and linear recurrence

Builds on Matrix Polynomials and the Minimal Polynomial and Invariant Subspaces and Block Matrices.

Video 42

generalized eigenvector, Jordan chain, nested kernels, stabilization of kernels, generalized eigenspace and independence of a chain

Builds on Matrix Polynomials and the Minimal Polynomial, Cyclic Subspaces and Companion Matrices and Invariant Subspaces and Block Matrices.

Video 43

primary decomposition, direct sum of generalized eigenspaces, nilpotent operator, split minimal polynomial and dimension of a generalized eigenspace

Builds on Polynomial Projectors and Diagonalizability, Generalized Eigenvectors and Direct Sums and Projections.

Video 44

Jordan block, Jordan form, Jordan basis, lifting a vector to a chain, top-down basis construction and dimension count for spanning

Builds on Generalized Eigenspace Decomposition, Generalized Eigenvectors and Basis Exchange and Dimension.

Video 45

successive kernel dimensions, counting Jordan blocks, uniqueness of block sizes, computing a Jordan basis and sensitivity of Jordan form to perturbation

Builds on Constructing Jordan Form and The Cayley-Hamilton Theorem.

Video 46

induced matrix norm, submultiplicativity, spectral norm and the largest singular value, norm of the adjoint, condition number, perturbation bound and effect of scaling on the condition number

Builds on The Singular Value Decomposition, Rayleigh Quotients and The Eckart-Young Theorem.

Video 47

residual, forward error, backward error, backward stability, residual bound on the forward error, ill-conditioned system, numerical rank and floating-point rounding model

Builds on Matrix Norms and Condition Numbers and The Pseudoinverse and Four Projectors. Also uses rounding a number to a fixed number of significant digits.

Video 48

Householder reflector, stable sign choice, Householder QR factorization, triangular solve, full and reduced QR and least-squares residual from QR

Builds on Orthogonal Projection and Best Approximation, Matrix Norms and Condition Numbers and Cauchy-Schwarz and Orthonormal Expansions.

Video 49

condition number of a rectangular matrix, conditioning of the Gram matrix, normal equations, Cholesky solve, partial pivoting and rank-aware method choice

Builds on Householder QR Factorization, Residuals, Forward Error and Backward Error and Positive Definite Matrices and Cholesky Factorization.

Video 50

power iteration, dominant eigenvalue, convergence rate of power iteration, starting-vector condition, equal-modulus eigenvalues, eigenpair residual and residual bound for normal matrices

Builds on The Spectral Theorem for Normal Operators, Computing Jordan Form and Its Limitations, Rayleigh Quotients and Residuals, Forward Error and Backward Error.

Video 51

polynomial function of a matrix, reduction modulo the minimal polynomial, powers of a Jordan block, binomial expansion with a nilpotent part and growth along a Jordan chain

Builds on The Cayley-Hamilton Theorem, Polynomial Projectors and Diagonalizability, Constructing Jordan Form and Cyclic Subspaces and Companion Matrices.

Video 52

exponential series, matrix exponential, convergence of the matrix series, exponential of a similar matrix, exponential of a sum of commuting matrices and transient growth

Builds on Matrix Powers and Polynomial Functions and Matrix Norms and Condition Numbers.

Video 53

linear update model, coupling matrix, eigenmode, mean and difference coordinates, mode decomposition, energy quadratic form and nonsymmetric coupling

Builds on Simultaneous Diagonalization, Quadratic Forms and Sylvester's Law of Inertia, The Min-Max Principle and The Matrix Exponential.

Video 54

inverse problem, noise amplification, truncated SVD reconstruction, Tikhonov regularization, filter factors, residual and solution norm tradeoff and unrecoverable component

Builds on The Moore-Penrose Conditions, Principal Component Analysis, Polar Decomposition, Residuals, Forward Error and Backward Error and QR, Normal Equations and Cholesky.

Video 55

dual of a quotient space, normality test, Jordan basis verification, orthogonal projector as a self-adjoint idempotent, forward difference operator and method choice under measurement noise

Builds on Coupled Systems and Eigenmodes, Inverse Problems and Regularization, Computing Jordan Form and Its Limitations, Adjoints and Gram Matrices and The First Isomorphism Theorem.