Algebra and geometry of lines, planes; solving simultaneously m equations in n unknowns; row reduction and row operations; reduced row echelon form; parametric representation of solutions, consistent vs. inconsistent systems, rank of a matrix. [June/July 2011 AMS Notices article: Mathematicians of Gaussian Elimination] [Online RREF Calculator]

For those who may need a refresher in vector operations, see Appendix A at the back of the Bretscher text or the referenced notes below. You may want to learn how to enter a matrix into a matrix-capable calculator and how to use the “rref” function to do row reduction.

Lecture #1 Notes     Lecture #1 Notes reformatted for Zoom

[Multivariable Calculus Lecture Notes #1 for basic vector references]

Product of matrix and vector, matrix form of a linear system. Linear transformation defined by a matrix; linearity property; meaning of the columns of a matrix. Identity matrix, dilation (scaling) matrix. Constructing matrices of common linear transformations in geometry - rotations, dilations, projections, reflections.

Supplement on the dot product and orthogonal projection
(for those who have not yet taken multivariable calculus)

Lecture #2 Notes     Lecture #2 Notes reformatted for Zoom

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

Algebraic and geometric properties of the dot product; projections and reflections. Invertible linear transformations and algorithm for finding the inverse of a matrix (when it exists). Matrix algebra, composition of linear functions and matrix products.

Lecture #3 Notes     Lecture #3 Notes reformatted for Zoom

Subspaces of Rⁿ, span of a collection of vectors; kernel and image of a linear transformation. Linear dependence and linear independence. Basis and dimension of a subspace; Rank-Nullity Theorem. Coordinates of a vector relative to a basis; examples.

Lecture #4 Notes     Lecture #4 Notes reformatted for Zoom

Matrix of a linear transformation relative to an alternate basis; interpretation of the columns of a matrix relative to a given basis; similarity of matrices. Introduction to general linear spaces (vector spaces), e.g. function spaces and families of matrices.

Lecture #5 Notes     Lecture #5 Notes reformatted for Zoom

Linear transformations from one vector space to another, isomorphisms, coordinates relative to a basis for a linear space or subspace. Image, kernel, rank, nullity of general linear transformations. Matrix of a linear transformation relative to bases for domain and codomain.

Lecture #6 Notes     Lecture #6 Notes reformatted for Zoom

Supplement on Representation of Functions in Different Coordinates

Inner product spaces; introduction to orthogonality, orthonormal bases, orthogonal complement of a subspace of Rⁿ; finding coordinates of a vector relative to an orthonormal basis; orthogonal projection; matrices for orthogonal projection and reflection using an ON basis.

Lecture #7-8 Notes    Lecture #7-8 Notes reformatted for Zoom

Inner product spaces; introduction to orthogonality, orthonormal bases, orthogonal complement of a subspace of Rⁿ; finding coordinates of a vector relative to an orthonormal basis; orthogonal projection; matrices for orthogonal projection and reflection using an ON basis.

Lecture #7-8 Notes    Lecture #7-8 Notes reformatted for Zoom

Exam #1 is scheduled to take place on Fri, July 7 - Sat, July 8 using Proctorio. There will be a 24-hour window starting at 12:00pm (noon) on Fri, July 7 and ending at 12:00pm on Sat, July 8 during which you must take the exam within Canvas/Proctorio. The exam will cover topics from Chapters 1-4 of the text (everything covered in class through Wed, July 6), but primarily Chapters 1-3 (and HW1-HW3). The exam should take approximately 70 minutes and additional time will be allotted for downloading, printing, scanning, and uploading (total time 90 minutes). Calculators are allowed unless otherwise stated, but no other notes or texts.

Practice Exam #1 (use same username/password as HW solutions)     Practice Exam #1 solutions (do the practice exam first!)     Exam #1 solutions

Most students will download and print the exam and do their work on the printed exam, but you may also simply view the exam within Proctorio (without printing it) and write up your exam solutions on paper (or a tablet) and then upload your work. Please be mindful to keep your file size reasonable and your work easily readable with sufficiently high contrast (black on white is best).

It is strongly recommended that you take the Proctorio Setup Quiz in Canvas (under the “Quizzes” tab) as soon as possible in order to make sure that your computer and Chrome browser are properly configured and that the Proctorio Extension is installed and working properly.

Facts about transposes. Orthogonal matrices. Gram-Schmidt process and QR factorization. Orthogonal transformations; orthogonal matrices. Method of Least Squares for approximate solutions to a linear system. Regression lines, data fitting, alternate method for finding matrix for orthogonal projection.

Lecture #7-8 Notes     Lecture #7-8 Notes reformatted for Zoom

Lecture #9 Notes     Lecture #9 Notes reformatted for Zoom

Supplement on Least Squares in Economics

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

Extra: Supplement on Elementary Row Operations, Row Spaces, and Reduced Row-Echelon Form (RREF) [includes discussion of the “Four Fundamental Subspaces” associated with any matrix A]

Last details of regression lines, data fitting, alternate method for finding matrix for orthogonal projection; Cobb-Douglas illustration of method. Introduction to Fourier approximation, Fourier coefficients; Fourier series calculations, and applications to infinite series.

Lecture #9 Notes (revised)     Lecture #9 Notes reformatted for Zoom

Lecture #10 Notes     Lecture #10 Notes reformatted for Zoom

Determinant of a square matrix, permutations, Laplace expansion, multilinearity, effect of row operations on the value of the determinant; matrix A invertible if and only if det(A) nonzero; det(AB) = det(A)det(B) and its corollaries; determinant of a linear transformation. Use of determinants to find area, volume, k-volume; determinant as an expansion factor; Cramer’s Rule and a not too useful formula for calculating the inverse of a matrix (see text).

Lecture #10 Notes     Lecture #10 Notes reformatted for Zoom

Lecture #11 Notes    Lecture #11 Notes reformatted for Zoom

Last details of determinants. Eigenvalues and eigenvectors of a (square) matrix; characteristic polynomial; finding eigenvalues and eigenvectors. Algebraic multiplicity (AM) vs. geometric multiplicity (GM) of an eigenvalue, diagonalization and criteria for the existence of a basis of eigenvectors. Discrete linear dynamical systems, powers of a matrix, and Markov matrix example. Distinct eigenvalues yield linearly independent eigenvectors; diagonalizability.

Lecture #12 Notes     Lecture #12 Notes reformatted for Zoom

Dealing with repeated eigenvalues where GM < AM. Stability of a discrete linear dynamical system in terms of the modulus of the eigenvalues. Final details on eigenvalues and the Jordan canonical form of a matrix. Spectral Theorem and orthogonal diagonalizability of symmetric matrices.

Lecture #14 Notes (for July 19 and July 21)

Lecture #14 Notes reformatted for Zoom

Exam #2 will take place beginning on Fri, July 21 using Proctorio within Canvas. There will be a 24-hour window during which you will be able to take the exam, scan your work, and submit the completed exam - just as we did for the first exam. Please do not wait until the last minute to do the exam and make sure you are fully familiar with the protocol for taking an exam in Proctorio. The exam will cover topics from Chapter 4 through Chapter 7, sections 7.1-7.4, i.e. through topics we cover on Monday (with a few added details on Wednesday).

Practice Exam #2     Solutions     Exam #2 solutions

Quadratic forms; positive definiteness of a matrix; Principal Axes Theorem; applications to ellipses, hyperbolas, ellipsoids and hyperboloids; 2nd derivative test for functions of several variables in terms of eigenvalues. Singular Values and Singular Value Decomposition.

Lecture #15 Notes     Lecture #15 Notes reformatted for Zoom

Vector fields; systems of linear ordinary differential equations and their solutions – case of real eigenvalues and diagonalizable coefficient matrix.

Lecture #16-17 Notes (some topics to be covered Wednesday)

Lecture #16-17 Notes reformatted for Zoom

Vector fields and continuous dynamical systems, continued - complex eigenvalue case; reduction of order; Hooke’s Law and oscillations, damped spring example; repeated eigenvalues, combination problems; stability, phase-plane analysis, analytic solutions. Linear differential operators and solutions to homogeneous and inhomogeneous linear differential equations, eigenfunctions, characteristic polynomial; kernel and image of a linear differential operator.

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

Lecture #18 Notes

Practice Final Exam     Solutions

Linear operators, continued. Nonlinear differential equations; nullclines, equilibria. Equilibrium analysis using Jacobian matrices; phase-plane analysis.

Lecture #18 Notes     Supplement on nonlinear systems

Final Exam Review at 9:30am