Math S-21b - Calendar of topics and HW assignments - Summer 2019

Last updated Friday, August 9, 2019 1:47 PM.
[This Calendar will be updated as the course progresses. Check back frequently.]

Date Topics Homework assignments
Mon,
June 24
week #1

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. 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

 HW #1: Read sections 1.1 through 1.3 of the Bretscher text and do the following problems: Due Fri, June 28: 1.1/1,3,7,17,25,29 1.2/10,16,30,34,36,42,70 1.3/2,3,4,26,47,48,56 For additional practice: 1.1/11,15,20,21 1.2/5,9,11,20-22,37,38,39 Chapter 1 True/False questions HW #1 problems (PDF)(problems keyed to the 4th Edition)
Wed,
June 26

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

 HW #2: Read sections 2.1 through 2.4 and Appendix A and do the following problems: Due by Wed, July 3: 2.1/43,44 2.2/6,7,19-23,34 2.3/56,58 2.4/2,4,10,12,66,67-75,     76,78,80,81 For additional practice: 2.1/5,6,24-30 2.2/4,5 2.3/3,4,11,12,55,57 2.4/41,49,50 Chapter 2 True/False questions HW #2 problems (PDF)
Fri,
June 28

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

Mon,
July 1
week #2

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.

Lecture #4 Notes

 HW #3: Read sections 3.1-3.4 and do the following problems: Due Mon, July 8: 3.1/11,20,32,34,39,44 3.2/24,36,40,48 3.3/24,30,32 3.4/6,18,26,28,44,46,50,56 For additional practice: 3.1/5,6,8,19,23,24,25,31,51 3.2/1,2,3,6,17,19,28,37,41,49 3.3/23,27,29,36,60,61,62,81 3.4/5,7,17,27,42,45,55 Chapter 3 True/False questions HW #3 problems (PDF)
Wed, July 3

Coordinates of a vector relative to a basis; examples. 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.

Lecture #5 Notes

Fri,
July 5

Introduction to general linear spaces (vector spaces), e.g. function spaces and families of matrices. 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.

Lecture #6 Notes

Supplement on Representation of Functions in Different Coordinates

Mon,
July 8
week #3

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

Practice Exam #1     Solutions

 HW #4: Read sections 4.1-4.3 and do the following problems: Due Fri, July 12: 4.1/20,26,30 4.2/6,25,52,53,66 4.3/14,22,27,28,44,47 For additional practice: 4.1/1,2,3,9,10,11,25,29 4.2/2,4,67,81 4.3/1,13 Chapter 4 True/False questions HW #4 problems (PDF)
Wed,
July 10

Introduction to orthogonality, orthonormal bases, orthogonal complement of a subspace of Rn. Facts about transposes. Orthogonal transformations; orthogonal matrices.

Lecture #7-8 Notes

Midterm Exam #1 (covering Chapters 1-4 of the text)

Exam #1 Solutions

Exam #1 will take place during the latter part of class on Wed, July 10. The exam will cover topics from Chapters 1-4 of the text (everything covered in class through Fri, July 5). The exam should take approximately 70 minutes.

 HW #5: Read sections 5.1-5.4 and the Supplement on Least Squares. Do problems: due Wed, July 17: 5.1/12,16,17,26 5.2/(14,28),34 5.3/31,40,42,44,46 5.4/4,5,6,7,10,16,20,22,32,38 For additional practice: 5.1/15,18,28,29 5.2/(6,20),(8,22),33,38,40,41 5.3/5-11,37,45,47 5.4/1,2,15,17,18,31,37,40,41,42 Chapter 5 T/F Problems HW #5 problems (PDF)
Fri,
July 12

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

Lecture #9 Notes (includes some topics to be covered on Monday)

Mon,
July 15
week #4

Inner product spaces; Fourier approximation, Fourier coefficients, Fourier series, and applications to infinite series.

Lecture #10 Notes (includes some topics covered Wednesday)

 HW #6: Read sections 5.5 and 6.1-6.3 and do the following problems: due Mon, July 22: 5.5/10,12,26,28 6.1/18,26,30,44 6.2/6,18,26,34 6.3/7,13,14,18,48 For extra practice: 5.5/3,4,9,19,20,27 6.1/9,16,17,34,43 6.2/5,9,17,25,40,41,43 6.3/3,19,20,23,24,49 Chapter 6 T/F problems HW #6 problems (PDF)
Wed,
July 17

Last details of inner product spaces. 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.

Lecture #11 Notes (includes some topics covered Friday)

Fri,
July 19

Determinants, continued: 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).

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.

Lecture #12 Notes (includes some topics to be covered on Monday)

 HW #7: Read sections 7.1-7.4 and do problems: Due Fri, July 26: 7.1/34,36,50 7.2/4,6,8,20,22,23,28 7.3/2,4,8,14,33,34,36,44 7.4/12,18,32,36,49,50 For extra practice: 7.1/1-6,15-21,33,35,39,53 7.2/5,7,15,21,24,25,26,27 7.3/11,16,21,27,35,41,47 7.4/4,5,6,11,19,47,48 HW #7 problems (PDF)

Practice Exam #2

Solutions

Mon,
July 22
week #5

Distinct eigenvalues yield linearly independent eigenvectors; diagonalizability. Phase portraits; similar matrices have the same characteristic polynomials, same eigenvalues, same algebraic and geometric multiplicities. Obstructions to diagonalizability (GM < AM, complex eigenvalues). Trace and determinant in terms of eigenvalues; characteristic polynomial and eigenvalues of a linear transformation. Review of complex numbers; complex eigenvalues and invariant (rotation-dilation) subspaces.

Lecture #13 Notes (some topics to be covered Wednesday)

Wed,
July 24

Complex eigenvalues and invariant (rotation-dilation) subspaces. Dealing with repeated eigenvalues where GM < AM.

Midterm Exam #2 (covering Chapters 4-6 and 7.1-7.4)     Solutions

 HW 8: Read sections 7.5-7.6 and 8.1-8.3 and the supplement on repeated eigenvalues, complex eigenvalues, and Jordan canonical form for a matrix. Optional: Read the supplement on quadratic forms, critical points, principal axes, and the 2nd derivative test. Due Wed, July 31: 7.5/20,24,28,30,32 7.6/10,12,24,38 8.1/10,12,16,19,20,24,36 8.2/4,6,8,16,18,22 For extra practice: 7.5/2,3,8,21,23,25,27,36 7.6/1-4,15 Chapter 7 T/F problems 8.1/3,5,6,15,29 8.2/3,9,11,15,19 Chapter 8 T/F problems HW #8 problems (PDF)
Fri,
July 26

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. Quadratic forms; positive definiteness of a matrix; principal axes; applications to ellipses, hyperbolas, ellipsoids and hyperboloids; 2nd derivative test for functions of several variables in terms of eigenvalues.

Lecture #15 Notes (new)

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.]
 HW #9: Read sections 9.1 through 9.3 and the Supplement on 1st order linear systems of differential equations and the use of evolution matrices. Also, read section 9.3 and the supplement on nonlinear systems. Do the following problems. Due Mon, Aug 5: 8.3/SVD Problem 9.1/24,(26,32),(28,34),31,52 (paired problems best done together) 9.2/7,12,31,36,39 9.3/4,24,28,44 For extra practice: 9.1/21,22,23,(29,35),     42,49,54,55 9.2/6,22-26,34 9.3/2,7,13,17,30,43 HW #9 problems (PDF)
Mon,
July 29
week #6

Continuation of quadratic forms and Principal Axes Theorem. Singular Values and Singular Value Decomposition. Vector fields; systems of linear ordinary differential equations and their solutions – case of real eigenvalues and diagonalizable coefficient matrix.

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

Wed,
July 31

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

 HW #10: Read the supplement on nonlinear systems. Do, but don't turn in: I: Problems 1-4 at the end of the nonlinear supplement. HW #10 problems (do, but don't turn in)

Practice Final Exam

Solutions to Practice Final Exam

Fri,
Aug 2

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

Supplement on nonlinear systems

Mon,
Aug 5
Final Exam Review at 9:30am in Emerson Hall 210
Fri,
Aug 9
FINAL EXAM (8:30am - 11:30am) in Emerson Hall 210 (our regular lecture hall)
There will be no alternate arrangements for this exam except for those with certified disabilities.