Linear Algebra I

Video 2

linear combination, coefficient, span, generating vectors, membership in a span and redundant generator

Builds on Vectors and Vector Operations.

Video 3

matrix, size of a matrix, matrix entry, matrix-vector product, product as a weighted sum of columns, matrix addition, scalar multiple of a matrix and transpose

Builds on Vectors and Vector Operations and Linear Combinations and Span.

Video 4

linear transformation, additivity, homogeneity, zero preservation, standard unit vectors, standard matrix of a linear transformation, translation and nonlinear map

Builds on Matrices and Matrix-Vector Multiplication. Also uses function notation for an input-output rule.

Video 5

composition of linear maps, matrix product, compatible sizes, noncommutativity of matrix multiplication, identity matrix, associativity, transpose of a product and block multiplication

Builds on Matrices and Matrix-Vector Multiplication and Linear Transformations.

Video 6

linear system, solution of a system, matrix form of a system, augmented matrix, row picture, column picture and consistent and inconsistent systems

Builds on Vectors and Vector Operations and Linear Transformations. Also uses graphing a straight line from its equation.

Video 7

Gaussian elimination, elementary row operations, row replacement, equivalent systems, elementary matrix, row echelon form and back-substitution

Builds on Matrix Multiplication and Composition and Linear Systems as Equations and Geometry.

Video 8

pivot, pivot column, free variable, reduced row echelon form, inconsistent row, parametric solution, particular solution and homogeneous solution

Builds on Gaussian Elimination and Row Operations and Linear Combinations and Span.

Video 9

coefficient parameter, family of linear systems, exceptional parameter value, case analysis, nonzero condition for division and parameter versus free variable

Builds on Gaussian Elimination and Row Operations, Linear Combinations and Span and Pivots, Free Variables and All Solutions.

Video 10

inverse matrix, invertible matrix, singular matrix, computing an inverse by row reduction, pivot test for invertibility and solving a system with an inverse

Builds on Matrix Multiplication and Composition, Gaussian Elimination and Row Operations and Linear Systems with Parameters.

Video 11

LU factorization, lower triangular matrix, upper triangular matrix, elimination multipliers, forward substitution, permutation matrix, pivoted LU factorization and partial pivoting

Builds on Matrix Multiplication and Composition, Gaussian Elimination and Row Operations and Inverse Matrices and Invertibility.

Video 12

classifying a system before solving, redundant equation, complete solution family, inconsistent right-hand side, exceptional coefficient, tracking a row swap and checking the original equations

Builds on Matrix Multiplication and Composition, Linear Systems as Equations and Geometry, Gaussian Elimination and Row Operations, Linear Systems with Parameters, Inverse Matrices and Invertibility and LU Factorization and Pivoting.

Video 13

subspace, subspace test, closure under addition and scaling, solution set of a homogeneous system, affine set, intersection of subspaces, sum of subspaces and union of subspaces

Builds on Linear Combinations and Span and Linear Systems with Parameters.

Video 14

linearly independent list, linearly dependent list, trivial combination, dependence relation, uniqueness of coefficients and independence test by elimination

Builds on Linear Combinations and Span, Gaussian Elimination and Row Operations and Subspaces, Sums and Intersections.

Video 15

basis, ordered basis, coordinate vector, pruning a spanning list, extending an independent list, dimension and standard basis

Builds on Linear Combinations and Span, Linear Systems with Parameters and Linear Independence.

Video 16

column space, row space, pivot columns of the original matrix, rank, equality of row rank and column rank and rank factorization

Builds on Linear Systems with Parameters, Subspaces, Sums and Intersections, Linear Independence and Bases and Dimension.

Video 17

null space, nullity, left null space, left nullity, four fundamental subspaces and ambient space of a subspace

Builds on Linear Systems with Parameters, Subspaces, Sums and Intersections, Linear Independence, Bases and Dimension and Column Space, Row Space and Rank.

Video 18

kernel, image, rank-nullity theorem, injective map, surjective map, fiber of a linear map and rank of a composition

Builds on Linear Transformations, Linear Systems with Parameters, Bases and Dimension and Null Space and Left Null Space.

Video 19

basis matrix, solving for basis coordinates, change-of-basis matrix, inverse change-of-basis matrix and coordinate change versus moving a vector

Builds on Matrix Multiplication and Composition, Inverse Matrices and Invertibility, Bases and Dimension and Kernel, Image and Rank-Nullity.

Video 20

matrix of a linear map in chosen bases, input and output bases, linear operator, similar matrices, commutative diagram and operator versus its matrix

Builds on Matrix Multiplication and Composition, Inverse Matrices and Invertibility, Bases and Dimension, Kernel, Image and Rank-Nullity and Change of Basis for Vector Coordinates.

Video 21

vector space, vector space axioms, polynomials of degree at most two, coordinates of a polynomial, space of 2-by-2 matrices and matrix units

Builds on Subspaces, Sums and Intersections, Linear Independence, Bases and Dimension, Kernel, Image and Rank-Nullity and Matrices in Different Bases. Also uses polynomials, their coefficients and degree.

Video 22

evaluation map, matrix of a map in a polynomial basis, kernel and image of an evaluation map, trace of a matrix and linear map determined by basis images

Builds on Subspaces, Sums and Intersections, Linear Independence, Bases and Dimension, Kernel, Image and Rank-Nullity, Matrices in Different Bases and Vector Spaces of Polynomials and Matrices.

Video 23

verifying a proposed basis, completeness by a dimension count, basis from the original columns, reconstructing a vector from its coordinates, changing the sample points of an evaluation map and graph of a polynomial versus its coordinates

Builds on Subspaces, Sums and Intersections, Linear Independence, Bases and Dimension, Null Space and Left Null Space, Kernel, Image and Rank-Nullity, Matrices in Different Bases and Linear Maps on Polynomials and Matrices.

Video 24

determinant, oriented area, orientation, 2-by-2 determinant formula, volume scale factor, orientation-reversing map and zero determinant and dependent columns

Builds on Linear Transformations, Matrix Multiplication and Composition, Inverse Matrices and Invertibility and Bases and Dimension. Also uses areas of rectangles, triangles and parallelograms.

Video 25

row operations and the determinant, determinant of a triangular matrix, minor, cofactor, cofactor expansion, linearity in one column and multiplicativity of the determinant

Builds on Gaussian Elimination and Row Operations, LU Factorization and Pivoting and Determinants as Oriented Area and Volume.

Video 26

determinant test for invertibility, singular square system, reachable and unreachable right-hand sides, Cramer's rule and Cramer's rule versus elimination

Builds on Inverse Matrices and Invertibility, Null Space and Left Null Space, Determinants as Oriented Area and Volume and Computing Determinants with Row Operations.

Video 27

dot product, Euclidean length, unit vector, distance between vectors, orthogonal vectors and angle between nonzero vectors

Builds on Vectors and Vector Operations, Bases and Dimension and Linear Maps on Polynomials and Matrices. Also uses the Pythagorean theorem and square roots.

Video 28

inner product, inner product axioms, weighted inner product, norm from an inner product, Cauchy-Schwarz inequality, equality case of Cauchy-Schwarz and triangle inequality

Builds on Vectors and Vector Operations, Bases and Dimension, Linear Maps on Polynomials and Matrices and Dot Products, Length and Angle. Also uses expanding the square of a sum or a difference.

Video 29

orthogonal list, orthonormal basis, projection onto a line, projection coefficient, residual vector, closest-point property and orthogonal matrix

Builds on Bases and Dimension and Inner Products and Cauchy-Schwarz.

Video 30

orthogonal complement, null space as the complement of the row space, left null space as the complement of the column space, complementary dimensions and orthogonal decomposition of a space

Builds on Null Space and Left Null Space, Kernel, Image and Rank-Nullity, Inner Products and Cauchy-Schwarz and Orthonormal Coordinates and Line Projection.

Video 31

orthogonal projection onto a subspace, normal equations, projection matrix, idempotence, closest vector in a subspace and basis independence of a projection

Builds on Inverse Matrices and Invertibility, Orthonormal Coordinates and Line Projection and Orthogonal Complements and the Four Subspaces.

Video 32

Gram-Schmidt process, orthogonal remainder, normalization, reduced QR factorization, matrix with orthonormal columns, upper triangular factor and zero remainder and dependence

Builds on LU Factorization and Pivoting, Linear Independence, Inner Products and Cauchy-Schwarz, Orthonormal Coordinates and Line Projection and Projection onto a Subspace.

Video 33

least-squares solution, sum of squared errors, residual vector in observation space, vertical residuals, normal equations for a least-squares fit and fitted vector

Builds on Linear Systems as Equations and Geometry, Orthogonal Complements and the Four Subspaces and Projection onto a Subspace. Also uses the equation of a straight line with an intercept and a slope.

Video 34

design matrix, model linear in its coefficients, fitted values, residual orthogonality check, dependent design columns and nonunique coefficients

Builds on LU Factorization and Pivoting, Gram-Schmidt and QR Factorization and Least Squares and Orthogonal Projection.

Video 35

least squares by QR factorization, triangular least-squares equation, orthonormal coordinates of the data, residual outside the model space and split of the squared error

Builds on LU Factorization and Pivoting, Gram-Schmidt and QR Factorization, Least Squares and Orthogonal Projection and Least-Squares Models and Normal Equations.

Video 36

perturbation of the data, absolute and relative error, induced matrix norm, condition number, ill-conditioned system, conditioning versus stability and residual versus solution error

Builds on Gaussian Elimination and Row Operations, Inverse Matrices and Invertibility, Inner Products and Cauchy-Schwarz, Gram-Schmidt and QR Factorization and Least Squares with QR.

Video 37

error criterion, projection onto a plane through its normal, line fit to observations, proportional design columns, sensitivity to a changed measurement and checking the model and residual

Builds on Inner Products and Cauchy-Schwarz, Orthonormal Coordinates and Line Projection, Orthogonal Complements and the Four Subspaces, Projection onto a Subspace, Gram-Schmidt and QR Factorization, Least Squares and Orthogonal Projection, Least Squares with QR and Conditioning and Sensitivity.

Video 38

eigenvector, eigenvalue, eigenpair, invariant line, negative eigenvalue, zero eigenvalue and independence of eigenvectors for distinct eigenvalues

Builds on Linear Transformations, Matrix Multiplication and Composition, Bases and Dimension and Inner Products and Cauchy-Schwarz.

Video 39

characteristic polynomial, characteristic equation, eigenspace, basis of an eigenspace and trace and determinant checks

Builds on Linear Systems with Parameters, Linear Independence, Computing Determinants with Row Operations, Determinants, Invertibility and Cramer's Rule and Eigenvectors and Eigenvalues. Also uses solving quadratic equations by factoring or the quadratic formula.

Video 40

repeated eigenvalue, algebraic multiplicity, geometric multiplicity, defective matrix and bound on geometric multiplicity

Builds on Linear Systems with Parameters, Linear Independence, Computing Determinants with Row Operations, Determinants, Invertibility and Cramer's Rule, Eigenvectors and Eigenvalues and Calculating Eigenvalues and Eigenspaces.

Video 41

diagonalization, eigenbasis, diagonalizable matrix, matrix powers in eigen-coordinates, diagonalizability test and repeated linear update

Builds on Matrix Multiplication and Composition, Inverse Matrices and Invertibility, Matrices in Different Bases and Algebraic and Geometric Multiplicity.

Video 42

imaginary unit, complex number, complex plane, complex conjugate, modulus, complex division and multiplication by i as a quarter-turn

Also uses arithmetic with fractions and squares and points and coordinates on a graph.

Video 43

complex eigenvalue, complex eigenvector, quarter-turn rotation matrix, conjugate pairs of eigenvalues, complex coordinate space, diagonalization over the complex numbers and conjugate transpose

Builds on Determinants as Oriented Area and Volume, Inner Products and Cauchy-Schwarz, Algebraic and Geometric Multiplicity, Diagonalization and Matrix Powers and Complex Numbers for Linear Algebra.

Video 44

symmetric matrix, real spectral theorem, orthogonal diagonalization, orthonormal eigenbasis, orthogonality of eigenvectors for distinct eigenvalues and orthonormal basis of a repeated eigenspace

Builds on Inner Products and Cauchy-Schwarz, Gram-Schmidt and QR Factorization, Diagonalization and Matrix Powers and Complex Eigenvalues and Real Rotations.

Video 45

quadratic form, matrix of a quadratic form, cross term, principal axes of a quadratic form, positive definite, negative definite, positive and negative semidefinite and indefinite form

Builds on Matrices in Different Bases, Inner Products and Cauchy-Schwarz, Diagonalization and Matrix Powers and The Spectral Theorem for Symmetric Matrices.

Video 46

choice of scalar field, test for a defective matrix, complex eigenpairs of a rotation, real orthogonal diagonalization, matrix power from eigenvalues and hypothesis of a method

Builds on Eigenvectors and Eigenvalues, Algebraic and Geometric Multiplicity, Diagonalization and Matrix Powers, Complex Eigenvalues and Real Rotations, The Spectral Theorem for Symmetric Matrices and Quadratic Forms and Positive Definiteness.

Video 47

singular value decomposition, singular values, right singular vectors, left singular vectors, largest stretch, rank from singular values and singular values versus eigenvalues

Builds on Null Space and Left Null Space, Kernel, Image and Rank-Nullity, Inner Products and Cauchy-Schwarz, Gram-Schmidt and QR Factorization, The Spectral Theorem for Symmetric Matrices and Quadratic Forms and Positive Definiteness.

Video 48

singular values from a symmetric eigenproblem, completing an orthonormal basis, full SVD, economy SVD, compact SVD, reconstruction check and repeated singular values

Builds on Null Space and Left Null Space, Kernel, Image and Rank-Nullity, Inner Products and Cauchy-Schwarz, Gram-Schmidt and QR Factorization, The Spectral Theorem for Symmetric Matrices, Quadratic Forms and Positive Definiteness and Singular Value Decomposition and Its Geometry.

Video 49

pseudoinverse, pseudoinverse of the diagonal factor, minimum-norm solution, least-squares solution of minimum norm and null-space component

Builds on Linear Systems with Parameters, Orthogonal Complements and the Four Subspaces, Least Squares and Orthogonal Projection and Computing an SVD in Full, Economy and Compact Forms.

Video 50

outer product, rank-one matrix, outer-product expansion of the SVD, truncated SVD, best rank-k approximation, Frobenius norm, spectral norm and tied singular values at the cutoff

Builds on Linear Maps on Polynomials and Matrices, Inner Products and Cauchy-Schwarz, Least Squares and Orthogonal Projection, Quadratic Forms and Positive Definiteness, Computing an SVD in Full, Economy and Compact Forms and Pseudoinverses and Minimum-Norm Solutions.

Video 51

feature mean, centered data matrix, sample covariance matrix, projected variance, principal component, retained variance, reconstruction error and effect of centering and units

Builds on Linear Maps on Polynomials and Matrices, Inner Products and Cauchy-Schwarz, Least Squares and Orthogonal Projection, Quadratic Forms and Positive Definiteness, Computing an SVD in Full, Economy and Compact Forms, Pseudoinverses and Minimum-Norm Solutions and Outer Products and Low-Rank Approximation.

Video 52

directed graph, incidence matrix, conservation at a node, net supply, circulation around a cycle, nonnegative flow constraint and total supply condition

Builds on Linear Systems as Equations and Geometry, Linear Systems with Parameters, Linear Independence, Null Space and Left Null Space and Kernel, Image and Rank-Nullity.

Video 53

overdetermined system, underdetermined system, residual check, minimum-norm reconstruction, effect of an added measurement and agreement with data versus a validated model

Builds on Linear Systems with Parameters, Null Space and Left Null Space, Matrices in Different Bases, Least Squares with QR, Conditioning and Sensitivity, Diagonalization and Matrix Powers, Computing an SVD in Full, Economy and Compact Forms, Pseudoinverses and Minimum-Norm Solutions and Principal Component Analysis with Centering and Variance.