Algebra and geometry of lines, planes; solving simultaneously m equations in n unknowns; consistent vs. inconsistent systems; row reduction and row operations; reduced row echelon form (RREF); parameterization of solutions; rank of a matrix; product of a matrix and a vector; vector and matrix forms of systems of linear equations.

Lecture #1 notes

[Multivariable Calculus Notes #1 for vector references]

Vector and matrix forms of systems of linear equations; linear transformations from R^m to Rⁿ defined by matrices; geometric meaning of linearity; domain and codomain; meaning of the columns of a matrix; rotations and dilations; shears; projections and reflections. Inverse of a linear transformation and method for finding the inverse of a (square) matrix.

Lecture #2 notes

[Multivariable Calculus Lecture Notes #2
for dot and cross product references]

Supplement on the dot product and orthogonal projection
(for those who did not take Math E-21a)

Inverse of a matrix; matrix algebra; associativity and composition of linear functions; image and kernel of a linear transformation; linear combinations and the span of a set of vectors; subspaces; linear independence; basis of a subspace.

Note: You may want to practice entering matrices on a calculator and performing matrix algebra with the calculator.

Lecture #3 notes

Test for linear independence, basis and dimension of a subspace, proof that dimension is well-defined; bases for kernels and images; Rank-Nullity Theorem; coordinates of a vector relative to a basis; matrix of a linear transformation relative to an alternate basis.

Lecture #4 notes

Matrix of a linear transformation relative to an alternate basis, applications to linear transformations defined geometrically.

General linear spaces, examples - continuous functions, differentiable functions, polynomials of degree less than or equal to n, the linear space of m by n matrices; complex numbers as a (real) 2-dimensional linear space. Subspaces of a linear space; basis and dimension; coordinates relative to a basis. General linear transformations and their matrices relative to a basis or bases; kernel, image, isomorphisms.

Lecture #5 notes

You should do the Proctorio Setup Quiz in Canvas (under Quizzes) to make sure that your Chrome browser is up-to-day and that the Proctorio Extension is properly installed and working. The exam will cover topics from the first five lectures of the course (Chapters 1-4 of the text).

Practice Exam #1 (use same username/password as HW solutions)

Solutions (do the Practice Exam first!)

A note on linearity in the solution of ordinary differential equations; orthogonality (perpendicularity) of vectors in Rⁿ; length (norm) of a vector, unit vectors; Cauchy-Schwarz inequality; orthogonal complements and method for finding them; introduction to orthogonal projections (see text and Lecture Notes for additional details).

Lecture #6 notes

Exam #1 in Proctorio during a 25-hour window Mar 6-7.

Exam #1 Solutions

Orthogonal projections; orthonormal basis; angle between two vectors; Gram-Schmidt orthogonalization process; QR factorization. Orthogonal transformations and orthogonal matrices. Least-squares approximation, normal equation; data-fitting.

Lecture #7 notes

Supplement on Least Squares in Economics

A caution in applying the Method of Least Squares
in regression analysis – Outliers

Extra Credit Problem – Four Fundamental Subspaces,
Pseudoinverses, Least Squares
After you have submitted HW7, if you’re still eager to do more, here’s an opportunity. This is purely optional. To receive any extra credit you must answer all questions completely. No partial credit. The deadline for submission is the same as HW7.

Determinant of a (square) matrix, patterns and permutations; Laplace expansion; multilinearity and the effect of the row operations on the value of the determinant; determinant criterion for invertibility of a matrix; k-volumes; determinant as an expansion factor; Cramer’s Rule and formula for finding A^-1 (minors, cofactors, and the classical adjoint).

Lecture #8 notes

Supplement on row operations, row spaces, and RREF

A different take on determinants by Sheldon Axler:
Down with Determinants!

Summary of facts about determinants and applications. Discrete (linear) dynamical systems, iteration of a matrix, trajectories and phase portraits; eigenvectors and eigenvalues of a (square) matrix; characteristic polynomial; algebraic and geometric multiplicities; diagonalization and the existence of a basis of eigenvectors; powers of a matrix.

Lecture #9 notes

Diagonalization and the existence of a basis of eigenvectors; powers of a matrix; trace and determinant; repeated eigenvalues; complex eigenvalues; review of facts about complex numbers; rotation-dilation matrices and complex eigenvalues.

Lecture #10 notes

Spectral Theorem; symmetric matrices and diagonalization by an orthonormal basis. Quadratic forms; positive definiteness of a matrix; principal axes; applications to ellipses and hyperbolas; Singular Values and Singular Value Decomposition (SVD).

Lecture #12 notes
(includes optional material on Hessian matrix and 2nd Derivative Test)

Exam #1 in Proctorio during a 25-hour window Apr 24-25 will cover chapters 4-7 of the text.

Exam #1 Solutions

EXTRA: Machine Learning – Singular Value Decomposition (SVD) & Principal Component Analysis (PCA)

Systems of linear differential equations and their solutions - distinct real eigenvalue case, complex eigenvalue case.

First order linear systems of differential equations and evolution matrices

Sampler of Phase Plane diagrams for uncoupled, coupled, period, and stable sink linear systems and a nonlinear system

Matrix Methods for Solving Systems of 1st Order Linear Differential Equations

Lecture #13 notes

Systems of linear differential equations and their solutions - complex eigenvalue examples, repeated eigenvalue case, and The Big Picture. Stability of equilibria. Nonlinear systems and linearization around equilibria.

Lecture #13 notes (used in Lecture #14)

Matrix Methods for Solving Systems of
1st Order Linear Differential Equations

Notes on Nonlinear Systems and Linearization

HW #14: Read sections 9.1 and 9.2 and the Notes on Nonlinear Systems and Linearization. There are five problems in the nonlinear supplement and solutions are posted for those problems.

HW #14 problems (PDF) - Do these, but don’t turn them in. Solutions will be posted.
[Derived from 9.2/6,7,12,36 and two problems from the Nonlinear Supplement]

Practice Final Exam     Solutions
(do the Practice Exam before consulting the solutions)

The two-hour Final Exam will take place Thursday, May 15 in Canvas/Proctorio (plus an extra 15 minutes for the usual downloading, printing, scanning, and uploading). Total time 135 minutes. The exam window will open at 12:00am and close at 11:59pm on Thursday.

No alternate exam dates will be permitted except as approved by Harvard Extension School Office.

The exam will cover topics from throughout the course with added emphasis on more recent topics not covered on previous exams.

Calculators will be permitted, but no other software or notes. Non-credit students are not permitted to take the Final Exam.