Math S-21b - Calendar of topics and HW assignments
Last updated 22 Aug 2005 04:01 PM .

Date Topics Text sections and 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. HW #1: Read section 1.1 and 1.2 of the Bretscher text and do the following problems (for class, not to be turned in):
1.1/1,3,7,15,17,29 and 1.2/5,11
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.
Tues, June 28  Last details of linear systems - row reduction methods, representation of solutions, consistent vs. inconsistent systems, rank of a matrix.  HW #2: Read sections 1.2 and 1.3 and do problems 1.2/10,16,30,34,36,42 and 1.3/2,3,4,24,26,47,48,56. These problems are to be submitted in class on Thursday. For those who may need a refresher in vector operations, see Appendix A at the back of the Bretscher text.
Wed, June 29  Product of matrix and vector, matrix form of a linear system. Linear transformation defined by a matrix; linearity property. HW #3: Read section 1.3 and parts of 2.1 and 2.2. Do (but don't turn in) problems 2.1/5,6,24-30 and 2.2/1,4,5 and as many of the Chapter 1 True/False questions as you can.
Thurs, June 30  Meaning of the columns of a matrix; identity matrix, dilation matrix. Algebraic and geometric properties of the dot product. Linear transformations in geometry - rotations, dilations, projections. HW #4: Read through section 2.3 (and 2.4 tomorrow) and Appendix A and do the following problems for Tues, July 5 (flexible due to holiday):
2.1/43,44 and 2.2/6,7,19-23,34 and
2.3/2,4,10,12,41 and 2.4/44,46.
Fri, July 1  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. HW #5: Read through 2.4 and do problems 2.4/3,4,11,12,15,16-25 and the Chapter 2 True/False questions on pgs. 94-95. These problems are for practice and are not to be turned in.
Mon, July 4 No class today - Independence Day (Monday Holiday)
Tues, July 5
week #2		 Finish up details of matrix algebra and geometrically defined linear transformations; subspaces of Rn, span of a collection of vectors; kernel and image of a linear transformation.
HW #6-7: Read sections 3.1 and 3.2 and do the following problems:
Due Fri, July 8:
2.4/48,49 
3.1/8,12,20,32,34,39,44 
3.2/18,24,28,36,40,48 		 For additional practice:
3.1/1,2,5,6,19,23-25,31 
3.2/1-3,6,17,19,37,41,49 
HW #6-7 problems (PDF)
Wed, July 6 Finding the kernel and image of a linear transformation; linear independence; basis for a subspace.
Thurs, July 7 Basis and dimension of a subspace; Rank-Nullity Theorem.
HW #8-9: Read sections 3.3 and 3.4 and do the following problems:
Due Tues, July 12:
3.3/24,30,32,36 
3.4/6,8,18,26,28,42,44,46,50,56 		 For additional practice:
3.3/23,27,29,40,41,42,61 
3.4/5,7,17,27,45,55 
Chapter 3 True/False questions 
HW #8-9 problems (PDF)        Practice Exam #1
Friday, July 8 Coordinates of a vector 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.
Mon, July 11 
week #3		 Matrix relative to a basis, continued. Introduction to general linear spaces, e.g. function spaces and families of matrices.
HW #10-11: 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 
4.3/22,27,28,47 		 For additional practice:
4.1/1-3,9,10,11,25,29 
4.2/2,4,66,67,81 
4.3/1,13,14,44 
Chapter 4 True/False questions 
HW #10-11 problems (PDF)
Tues, July 12 Linear spaces, continued. Linear transformations and isomorphisms, coordinates relative to a basis for a linear space or subspace. Image, kernel, rank, nullity of general linear transformations.
Wed, July 13

Midterm Exam #1

 HW #12: No new HW.
Thurs, July 14 Last details of general linear spaces: isomorphisms, examples. Dot product, orthogonality,  orthonormal bases and their advantages.
HW #13-14: Read sections 5.1-5.3 and do the following 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 		 For additional practice:
5.1/3,5,15,18,29 
5.2/6,20,33,40,41 
5.3/31,37,45,47 
HW #13-14 problems (PDF)
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; orthogonal matrices; Gram-Schmidt orthogonalization process.
Mon, July 18 
week #4		 Gram-Schmidt process and QR factorization, continued; orthogonal transformations; orthogonal matrices, using determinants to find area, volume, k-volume.
HW #15-17: Read sections 5.3, 5.4, and 5.5 and the Supplement on Least Squares. Do problems (by Fri, July 22):
Due Fri, July 22:
5.4/4,5,6,7,10,16,20,22,32,38 
5.5/4,10,20,26,28 		 For extra practice:
5.4/1,2,15,17,18,31,37,40,41,42 
5.5/3,9,12,19,27 
Chapter 5 T/F Problems 

HW #15-17 Problems (PDF)
Tues, July 19 Method of Least Squares, approximate solutions, regression lines, and data fitting; alternate method for finding matrix for orthogonal projection.
Wed, July 20 Last details of Least Squares. Inner product spaces; examples.
Thurs, July 21 Fourier approximation, Fourier coefficients, Fourier series, and applications to infinite series. HW #18: Read section 6.1 on determinants. Next HW below.
Fri, July 22 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.
HW #19-20: Read chapter 6 and do the following problems: 
To be turned in by Wed, July 27:
6.1/18,26,30,34,44 
6.2/6,18,26,34 
6.3/7,13,14,18,24 		 For extra practice:
6.1/9,16,17,43 
6.2/5,9,17,25,40,41,43 
6.3/3,19,20,23,48,49 
Chapter 6 T/F problems 

HW #19-20 Problems (PDF)
Mon, July 25 
week #5		 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. HW #19-20: See above.
Tues, July 26 Introduction to discrete dynamical systems and the eigenvalues and eigenvectors of a (square) matrix; characteristic polynomial; finding the eigenvalues and eigenvectors of a matrix; algebraic multiplicity (AM) and geometric multiplicity (GM) of an eigenvalue; Markov matrix example.
HW #21-23: Read sections 7.1-7.4 and do problems:
To be turned in Mon, Aug 1: 
7.1/33,34,35,36 
7.2/4,6,8,20,28 
7.3/2,4,8,14,33,44 
 		 For extra practice:
7.1/1-6,15-21,39,53 
7.2/5,7,15,21-27 
7.3/11,16,21,27,34-36,41,47 
7.4/4-6,11,19,47,49 

 HW #21-23 Problems (PDF)

Wed, July 27 Eigenvalues and eigenvectors, continued. Diagonalization and the existence of a basis of eigenvectors; powers of a matrix; criteria for a matrix to be diagonalizable;  phase portraits.
Thurs, July 28 Eigenvalues of a linear transformation; trace and determinant in terms of eigenvalues; how to deal with repeated eigenvalues where GM < AM.
Fri, July 29

Review of complex numbers. Complex eigenvalues and invariant (rotation-dilation) subspaces. Stability of a discrete linear dynamical system.

HW #24: Read sections 7.5 and 7.6 and study for the exam.
 Exam Topics and Practice Exam #2   

Mon, Aug 1 
week #6		 Midterm Exam #2 
HW #25-27: Read sections 7.5, 7.6, 8.1 and the supplement on repeated eigenvalues, complex eigenvalues, and Jordan canonical form for a matrix.

Due Fri, Aug 5 or Mon, Aug 8:
7.5/20,24,28,30,32 
7.6/10,12,18,24,38 
8.1/6,10,12,16,20,24,36 

 For extra practice:
7.5/21,22,27,29,36 
7.6/1-4,17,37,40 
Chapter 7 T/F problems
8.1/3,5,15,19,29 

HW #25-27 Problems (PDF)

Tues, Aug 2 Complex eigenvalues and invariant subspaces; rotation-dilation matrices; stability of a discrete linear dynamical system. Spectral Theorem.
Wed, Aug 3 Spectral Theorem and orthogonal diagonalizability of symmetric matrices.
Thurs, Aug 4 Quadratic forms; positive definiteness of a matrix; principal axes; applications to ellipses and hyperbolas; 2nd derivative test for functions of several variables in terms of eigenvalues (optional).
HW #28-29:: Read sections 8.2 and 9.1 and the supplement on quadratic forms, critical points, principal axes, and the 2nd derivative test.
Due Wed, Aug 10:
8.2/4,6,8,16,18,22 
9.1/24,(26,32),(28,34),(29,35),31,42,52 
(paired problems best done together) 		 For extra practice: 
8.2/1,2,3,9,11,15,19,21 
Chapter 8 True/False 
9.1/4,5,13,21,22,23,43,49,54,55 
Exercises at end of supplement on 2nd derivative test.

HW #28-29 Problems (PDF)
Fri, Aug 5 Systems of linear differential equations and their solutions – case of real eigenvalues and diagonalizable coefficient matrix.
Mon, Aug 8 
week #7		 Systems of linear differential equations, continued - complex eigenvalue and repeated eigenvalue cases; reduction of order; Hooke's Law and oscillations, damped spring example.

Here's a website that has a good java-based tool for doing phase-plane analysis: http://math.rice.edu/~dfield/dfpp.html. Contrary to what it says on this page, you do not need MATLAB or 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 #30-32: Read sections 9.2 and the supplement on nonlinear systems, and  do the following problems:
Due Fri, Aug 12 or Mon, Aug 15 
I:   9.2/6,7,12,31,36 and the following problem:
II: a) Find the general solution for the following system of differential equations:   
               
    b) Find the solution in the case where  
        x(0) = (5, 4, 3, 2, 1).    Solution
III: Do problems 1,2,4 at the end of the supplement. 		 For extra practice:
9.2/22-26,34,39 and 
Problems 3 and 5 at the end of the supplement.

HW #30-32 Problems (PDF)
Tues, Aug 9 Repeated eigenvalues, combination problems. Introduction to nonlinear differential equations; behavior near an equilibrium; phase-plane analysis. 
Wed, Aug 10 Introduction to nonlinear differential equations; behavior near an equilibrium; phase-plane analysis. 
Thurs, Aug 11 More example of phase-plane analysis of nonlinear systems of differential equations. Linear differential operators, linear differential equations, and solutions to homogeneous and inhomogeneous linear differential equations.  HW #33: Read section 9.3 and do problems 9.3/2,4,7,13,17,24,28,30,43 but don't turn them in.
Fri, Aug 12 Differential operators, continued:  Eigenfunctions, characteristic polynomials; kernel and image of a linear differential operator.  HW #34: Try the Practice Final Exam.
Mon, Aug 15

Final Exam Review at 11:00am in Science Center 222.

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Wed, Aug 17

FINAL EXAM - Wed, Aug 17, 9:00am in Science Center Hall C
This is a 3 hour exam and the time limit is strictly enforced.

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