Math E-21b – Spring 2020 – Calendar of topics and HW assignments
Last updated Friday, February 14, 2020 8:15 AM

Date Topics Text sections and homework assignments [Solutions]
(Check back after class each week for possible changes)
Underlined problems indicate those selected for grading.
Thurs, Jan 30
(Class #1)

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]

 HW #1: Read Chapter 1 of the Bretscher text (sections 1.1 through 1.3) and do the following problems for next week: To be turned in Feb 6 in class or online by Sat, Feb 8: 1.1/11,16,17,25,29 1.2/9,11,20,21,22,30,42,70 1.3/22,23,47,48 For additional practice: Chap 1 T/F questions If you're interested in economics, you may want to try 1.1/20,21  and  1.2/37-39. HW #1 problems (PDF) - keyed to 4th Edition You will probably want to make use of a calculator with the “rref” function and learn about how to enter a matrix into a calculator.
Thurs, Feb 6
(Class #2)
Vector and matrix forms of systems of linear equations; linear transformations from Rm 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.

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

Lecture #2 notes

 HW #2: Read sections 2.1-2.4 of Bretscher and do problems: To be turned in Feb 13 in class or online by Sat, Feb 15: 2.1/8,24-30,44 2.2/6,7,10,11,19,20,22,23,34 2.4/2,4,6,20,54 For additional practice: 2.1/5,7,9,37,43 2.2/4,5,17,21,24,27,28 2.4/40,41,42,43,53 2.4/49,50 (economics) HW #2 problems (PDF) - keyed to 4th Edition If you don't know much about vectors, the dot product, and the cross products (in R3), read Appendix A in the back of the text or, Multivariable Calculus Lecture Notes #2.
Thurs, Feb 13
(Class #3)

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

 HW #3: Read sections 2.3-2.4, 3.1-3.2 of Bretscher and do problems: To be turned in Feb 20 in class or online by Sat, Feb 22: 2.3/14 2.4/66,67-75,76,78,80,81 3.1/6,8,12,20,22,32,34,39,44 For additional practice: 2.3/1,2,3,4,7,10,11,12,27 3.1/1,2,5,19,23-25,31 2.4/101-103 (economics) Chapter 2 T/F questions HW #3 problems (PDF) - keyed to 4th Edition
Thurs, Feb 20
(Class #4)

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

 HW #4: Read sections 3.1-3.4 of Bretscher and do problems: To be turned in Feb 27 in class or online by Sat, Feb 29: 3.2/18,28,36,40,48 3.3/24,30,32,36 3.4/6,8,26,28,34,42,44,46,50,56,58 For additional practice: 3.2/1-3,6,17,19,24,37,41,49 3.3/23,27,29,60-62,81 3.4/5,7,17-18,27,32,33,37,39,45,55 Chapter 3 T/F questions HW #4 problems (PDF) - keyed to 4th Edition
Thurs, Feb 27
(Class #5)

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

Practice Exam #1     Solutions

 HW #5: Read sections 3.4 and 4.1-4.3, and do problems: To be turned in Mar 5: 3.4/60,62,70,72,73,74 4.1/20,26,30 4.2/6,25,52,53 4.3/22,27,28,47 For additional practice: 3.4/59,69,71 4.1/1-3,9,10,11,25,29 4.2/2,4,66,67,81 4.3/1,13,14,44 Chapter 4 T/F Problems HW #5 problems (PDF) - keyed to 4th Edition Note: Some of the details in Chapter 4 will be left to the reading. What you should get out of this chapter and the exercises is the sense that most constructions in Rn have analogues in the context of more general linear spaces and transformations.
Thurs, Mar 5
(Class #6)

Orthogonality (perpendicularity) of vectors in Rn; length (norm) of a vector, unit vectors; Cauchy-Schwartz 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     Solutions

 HW #6: Read sections 5.1, 5.2, and the first page of 5.4, and do problems: To be turned in Mar 12: 5.1/12,15,16,17,18,26,28 5.4/2,4,5,16 For additional practice: 5.1/3,5,23,29 5.4/1 HW #6 problems (PDF) - keyed to 4th Edition
Mon, Mar 12
(Class #7)

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

 HW #7: Read sections 5.2-5.4 and the Supplement on Least Squares in Economics and do problems: To be turned in Mar 26: 5.2/8,14,22,28,34,45 5.3/5-11,31,32,40,42,44,46,48,68 5.4/6,7,10,22,26,32,38,41,42 For extra practice: 5.2/6,20,33,44 5.3/45,47 5.4/15,17,18,24,37 Chapter 5 T/F Problems HW #7 problems (PDF) - keyed to 4th Edition If you have some free time, you may want to look over section 5.5 on inner product spaces, a generalization of the dot product in the context of general linear spaces. This is particularly useful in understanding Fourier series and Quantum Mechanics. [Optional Notes on Inner Product Spaces and Fourier Series]
Thurs, Mar 19 No Class - Harvard Spring Break
Thurs, Mar 26
(Class #8)

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

 HW #8: Read sections 6.1 to 6.3 and do problems: To be turned in Apr 2: 6.1/18,26,30,34,44 6.2/6,17,18,25,26,34,40 6.3/7,13,14,18,24 For extra practice: 6.1/9,16,17,43 6.2/5,9,41,43 6.3/3,19,20,23,48 Chapter 6 T/F problems HW #8 problems (PDF) - keyed to 4th Edition
Thurs, Apr 2
(Class #9)

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

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

 HW #9: Read sections 7.1-7.4 and do problems: To be turned in Apr 9 in class or online by Sat, Apr 11, 11:59pm EST: 7.1/33,34,35,36,50 7.2/4,6,8,20,21,22,23,24,25,26,27,28    7.3/2,4,8,14 For extra practice: 7.1/1-6,15-21,39,53 7.2/5,7,15 7.3/11,16,21 HW #9 problems (PDF) - keyed to 4th Edition
Thurs, Apr 9
(Class #10)

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

 HW #10: Read sections 7.4-7.5 and supplement on repeated eigenvalues, complex eigenvalues and Jordan canonical form and do problems: To be turned in Apr 16: 7.2/40,41 7.3/32,34,36,44 7.4/6,12,18,19,32,36,48,50 7.5/20,24,28,30,32 For extra practice: 7.3/27,33,35,47 7.4/4,5,11,47,49 7.5/21,22,27,29,36 Chapter 7 T/F problems Note: The matrix in problem 7.5/36 is known as a Leslie matrix. It incorporates birth rates and survival rates by age group for a given population. HW #10 problems (PDF) - keyed to 4th Edition
Thurs, Apr 16
(Class #11)

Rotation-dilation matrices and complex eigenvalues; eigenvalues and stability of a discrete linear dynamical system (phase portraits). Examples of complex and repeated eigenvalue cases. Spectral Theorem; symmetric matrices and diagonalization by an orthonormal basis.

Lecture #11 notes

Practice Exam #2     Solutions

 HW #11: Read sections 7.5-7.6 and supplement on repeated eigenvalues, complex eigenvalues and Jordan canonical form, and section 8.1 and do the following problems: To be turned in Apr 23: 7.6/10,12,17,18,24,38 8.1/6,10,12,16,24 For extra practice: 7.6/1-4,37,40 8.1/3,15 HW #11 problems (PDF) - keyed to 4th Edition
Thurs, Apr 23
(Class #12)

Spectral Theorem; symmetric matrices and diagonalization by an orthonormal basis. Quadratic forms; positive definiteness of a matrix; principal axes; applications to ellipses and hyperbolas.

Exam 2     Solutions

The exam will cover topics from Chapers 4-7 of the text.

 HW #12: Read sections 8.1-8.2 and the supplement on the Hessian matrix and the 2nd derivative test (optional) and do the following problems: To be turned in Apr 30: 8.1/5,19,20,29,36 8.2/4,6,8,9,11,16,18,19,22 For extra practice and enlightenment: 8.2/1,2,3,15,21 Chapter 8 T/F problems Problems #1 and #2 from supplement HW #12 problems (PDF) - keyed to 4th Edition
Here's a website that has a good java-based tool for showing vector fields and flows: http://math.rice.edu/~dfield/dfpp.html. The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.]
Thurs, Apr 30
(Class #13)

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

 HW #13: Read sections 9.1 and 9.2 and the supplement on 1st order linear systems of differential equations and the use of evolution matrices. Do the following problems: To be turned in May 7: 9.1/24,(26,32),(28,34),(29,35),31,42,52   (do paired problems together) 9.2/6,7,12,22-26,31,36,39 For extra practice and enlightenment: 9.1/4,5,13,21,22,23,43,49,54,55 9.2/34 HW #13 problems (PDF) - keyed to 4th Edition
Thurs, May 7
(Class #14)

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

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

Supplement on nonlinear systems and linearization

HW #14: Read sections 9.1 and 9.2 and the supplement on nonlinear systems and linearization. There are five problems in the nonlinear supplement and solutions are posted for those problems.

Practice Final Exam    Solutions

Thurs, May 14 Two-hour FINAL EXAM from 8:00pm to 10:00pm at a location to be determined
(No alternate exam dates or times will be permitted except via approved petition through the Harvard Extension School office)