Math S-21b - Calendar of topics and HW assignments - Summer 2011

Last updated Saturday, August 27, 2011 8:17 PM.

Date Topics Homework assignments
Mon,
June 27 
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]

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 Fri, July 1:
1.1/1,3,7,11,15,17,25,29
1.2/9,10,16,20-22,30,34,36,42,70
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 (PDF)
(problems are keyed to the 4th Edition)
Wed,
June 29

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)

HW #2: Read sections 2.1 through 2.4 and Appendix A and do the following problems:
Due Wed, July 6:
2.1/43,44
2.2/6,7,19-23,34
2.3/55,56,57,58
2.4/2,4,10,12,41,66,67-75,76,78,80,81 		For additional practice:
2.1/5,6,24-30
2.2/1,4,5
2.3/3,4,11,12
2.4/49,50
Chapter 2 True/False questions

HW #2 problems (PDF)
(problems are keyed to the 4th Edition)
Fri,
July 1		 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,
July 4
week #2		 Independence Day - holiday
HW #3: Read sections 3.1-3.4 and do the following problems:
Due Mon, July 11:
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
3.3/23,27,29,60,61,62,81
3.4/5,7,17,27,45,55
Chapter 3 True/False questions
HW #3 problems (PDF)
(problems are keyed to the 4th Edition)
Wed,
July 6		 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.
Fri,
July 8		 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.  Practice Exam #1
HW #4: Read sections 4.1-4.3 and do the following problems:
Due Fri, July 15:
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)
(problems are keyed to the 4th Edition)
Mon,
July 11 
week #3		 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.
Wed,
July 13

Examples of general linear spaces and related ideas. Dot product, orthogonality, orthonormal bases, orthogonal complement of a subspace of Rn.

Midterm Exam #1
HW #5: Read sections 5.1-5.5 and the Supplement on Least Squares. Do problems:
Due Wed, July 20:
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 (PDF)
(problems are keyed to the 4th Edition)
Fri,
July 15		 Finding coordinates of a vector relative to an orthonormal basis; orthogonal projection; matrices for orthogonal projection and reflection using an ON basis.Gram-Schmidt process and QR factorization. Orthogonal transformations; orthogonal matrices; facts about transposes. Method of Least Squares for approximate solutions to a linear system.
Mon,
July 18
week #4		 Least squares solutions, regression lines, data fitting, 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 Mon, July 25:
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 (PDF)
(problems are keyed to the 4th Edition)
Wed,
July 20		 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.
Fri,
July 22

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. Distinct eigenvalues yield linearly independent eigenvectors.  Practice Exam #2
HW #7: Read sections 7.1-7.4 and do problems:
Due Fri, July 29:
7.1/33,34,35,36,50
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:
7.1/1-6,15-21,39,53
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)
(problems are keyed to the 4th Edition)
Mon,
July 25
week #5		 Discrete linear dynamical systems, powers of a matrix, and Markov matrix example. 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.
Wed,
July 27

Complex eigenvalue case, continued. Repeated (real) eigenvalue case. Stability of a discrete linear dynamical system in terms of the modulus of the eigenvalues.

Midterm Exam #2
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 Wed, Aug 3:
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
7.6/1-4,15,17
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)
(problems are keyed to the 4th Edition)
Fri,
July 29

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.

Supplement on quadratic forms, critical points, principal axes, and the 2nd derivative test
  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.
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 8:
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 (PDF)
(problems are keyed to the 4th Edition)
Mon,
Aug 1
week #6

Last details of quadratic forms and Principal Axes Theorem. Vector fields; systems of linear ordinary differential equations and their solutions – case of real eigenvalues and diagonalizable coefficient matrix. Complex eigenvalue case; reduction of order; Hooke's Law and oscillations, damped spring example.
Wed,
Aug 3

Vector fields and continuous dynamical systems, continued - 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.
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,
Aug 5

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

Supplement on nonlinear systems
Mon,
Aug 8

Final Exam Review, 9:30am to 11:30am in our regular classroom - Harvard 202.

Practice Final Exam
Fri,
Aug 12

FINAL EXAM (8:30am - 11:30am) in Harvard Hall 202

Return to main Math S-21b page