Math S-21b - Calendar of topics and HW assignments - Summer 2010
Last updated 15 Mar 2010 08:56 PM.  Dates, topics, and assignments will change as the course proceeds.

Date Topics Homework assignments
Mon, June 21 
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.

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.

HW #1: Read sections 1.1 through 1.3 of the Bretscher text and do the following problems:

Due ---:
1.1/1,3,7,11,15,17,25,29
1.2/9,10,16,20-22,30,34,36,42
1.3/2,3,4,24,26,47,48,56		 For additional practice:
1.1/20,21
1.2/5,11,37,38,39
Chapter 1 True/False questions

HW #1 problems
Wed, June 23

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.

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

Identity matrix, dilation (scaling) matrix. Constructing matrices of common linear transformations in geometry - rotations, dilations, projections, reflections.
HW #2: Read sections 2.1 through 2.4 and Appendix A and do the following problems:
Due ---:
2.1/43,44
2.2/6,7,19-23,34
2.3/2,4,10,12,41
2.4/16-25,34,36,44,46,48,49		 For additional practice:
2.1/5,6,24-30
2.2/1,4,5
2.3/49,50
2.4/3,4,11,12,15
Chapter 2 True/False questions

HW #2 problems
Fri, June 25 Algebraic and geometric properties of the dot product. Invertible linear transformations and algorithm for finding the inverse of a matrix (when it exists). Matrix algebra, composition of linear functions and matrix products. Subspaces of Rn, span of a collection of vectors; kernel and image of a linear transformation.
Mon, June 28
week #2		 Finding the 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.
HW #3: Read sections 3.1-3.4 and do the following problems:
Due ---:
3.1/12,20,32,34,39,44
3.2/24,36,40,48
3.3/24,30,32,36
3.4/6,18,26,28,42,44,46,50,56		 For additional practice:
3.1/5,6,8,19,23,24,25,31
3.2/1,2,3,6,17,19,28,37,41,49
3.3/23,27,29,40,41,42,61
3.4/5,7,17,27,45,55
Chapter 3 True/False questions

HW #3 problems
Wed, June 30

Examples of coordinates relative to a basis; 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.
Fri, July 2 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.
HW #4: Read sections 4.1-4.3 and do the following problems:
Due ---:
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
Mon, July 5 
week #3

Independence Day - holiday
Wed, July 7

Dot product, orthogonality, orthonormal bases and their advantages.  Finding coordinates of a vector relative to an orthonormal basis; orthogonal projection; matrices for orthogonal projection and reflection using an ON basis; orthogonal matrices. Gram-Schmidt process.

Midterm Exam #1 covering Chapters 1 to 3 (date tentative).
HW #5: Read sections 5.1-5.5 and the Supplement on Least Squares. Do problems:
Due ---:
5.1/12,16,17,26,28
5.2/8,14,22,28,34,38
5.3/5-11,40,42,44,46
5.4/4,5,6,7,10,16,20,22,32,38		 For additional practice:
5.1/15,18,29
5.2/6,20,33,40,41
5.3/31,37,45,47
5.4/1,2,15,17,18,31,37,40,41,42
Chapter 5 T/F Problems 
HW #5 problems
Fri, July 9 Gram-Schmidt process and QR factorization. Orthogonal transformations; orthogonal matrices; facts about transposes; Method of Least Squares for approximate solutions to a linear system, regression lines, and data fitting.
Mon, July 12
week #4		 Alternate method for finding matrix for orthogonal projection. Inner product spaces; Fourier approximation, Fourier coefficients, Fourier series, and applications to infinite series.
HW #6: Read chapter 6 and do the following problems: 
Due ---:
5.5/4,10,20,26,28
6.1/18,26,30,34,44
6.2/6,18,26,34
6.3/7,13,14,18,24,48,49		 For extra practice:
5.5/3,9,12,19,27
6.1/9,16,17,43
6.2/5,9,17,25,40,41,43
6.3/3,19,20,23
Chapter 6 T/F problems
HW #6 problems
Wed, July 14 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.
Fri, July 16

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. Distinct eigenvalues yield linearly independent eigenvectors. Discrete linear dynamical systems, powers of a matrix, and Markov matrix example.
HW #7: Read sections 7.1-7.4 and do problems:
Due ---:
7.1/33,34,35,36,50
7.2/4,6,8,20,22,23,28
7.3/2,4,8,14,33,36,44
7.4/12,18,32,36,49,50		 For extra practice:
7.1/1-6,15-21,39,53
7.2/5,7,15,21,24,25,26,27
7.3/11,16,21,27,34,35,41,47
7.4/4,5,6,11,19,47,48
HW #7 problems
Mon, July 19
week #5		 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. Dealing with repeated eigenvalues where GM < AM. Review of complex numbers; complex eigenvalues and invariant (rotation-dilation) subspaces.
HW 8: Read sections 7.5 and 7.6 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 ---:
7.5/20,24,28,30,32
7.6/10,12,24,38
8.1/10,12,16,20,24,36
8.2/4,6,8,16,18,22

 For extra practice:
7.5/2,3,8,21,23,25,27,36
7.6/1-4,15-18
Chapter 7 T/F problems
8.1/3,5,6,15,19,29
8.2/1,2,3,9,11,15,19,21
Chapter 8 T/F problems
HW #8 problems
Wed, July 21

Complex eigenvalue case, continued. Stability of a discrete linear dynamical system. Spectral Theorem and orthogonal diagonalizability of symmetric matrices.

Midterm Exam #2 covering Chapters 4, 5, 6, and 7.1-7.4.
Fri, July 23

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. Introduction to systems of linear ordinary differential equations and their solutions – case of real eigenvalues and diagonalizable coefficient matrix; vector fields.
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 ---:
9.1/24,(26,32),(28,34),(29,35),31,42,52
(paired problems best done together)
9.2/6,7,12,31,36
PLUS the Big Problem (see PDF)
Problems 9.3/2,4,24,28,44		 For extra practice:
9.1/21,22,23,49,54,55
9.2/22-26,34,39
9.3/7,13,17,30,43.
HW #9 problems
  Here's a website that has a good java-based tool for doing phase-plane analysis: http://math.rice.edu/~dfield/dfpp.html. You do not need any other software to use this tool. Choose the PPLANE option. You can enter new functions and change the size of the window. To see trajectories, just click on a point in the phase-plane. You should be able to print the phase portraits produced by this tool.
Mon, July 26
week #6		 Systems of linear differential equations - complex eigenvalue case; reduction of order; Hooke's Law and oscillations, damped spring example. Repeated eigenvalues, combination problems; stability, phase-plane analysis, analytic solutions. Intro. to linear differential operators and solutions to homogeneous and inhomogeneous linear differential equations.
Wed, July 28 Linear differential operators, continued; eigenfunctions, characteristic polynomials; kernel and image of a linear differential operator. Nonlinear continuous dynamical systems; equilibrium analysis.
HW #10: Read the supplement on nonlinear systems.
Do, but don't turn in:
I: Problems 1,2,3, and 4 at the end of the nonlinear supplement.
HW #10 problems (do, but don't turn in)
Fri, July 30 Nonlinear differential equations; nullclines, equilibria. Equilibrium analysis using Jacobian matrices; phase-plane analysis. Supplement on nonlinear systems 
Aug 3, Aug 5

Final Exam Reviews

-
Fri, Aug 7 (?)

FINAL EXAM - actual date, time, and location to be determined

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