Math E-21b – Spring 2012 – Calendar of topics and HW assignments
Last updated Thursday, February 2, 2012 11:38 PM
This Calendar will change as the course proceeds.

Date Topics Text sections and homework assignments
(Check back after class each week for possible changes)
Thurs, Jan 26
(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.
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 2:
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 2
(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 [read about this in the text - we'll fill in some of the details next week].

Supplement on the dot product and orthogonal projection
(for those who did not take Math E-21a)
HW #2: Read sections 2.1-2.4 of Bretscher and do problems:
To be turned in Feb 9:
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, better yet, any multivariable calculus text.
Thurs, Feb 9
(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.
HW #3: Read sections 2.3-2.4, 3.1-3.2 of Bretscher and do problems:
To be turned in Feb 16:
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 16
(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.
HW #4: Read sections 3.2-3.4 of Bretscher and do problems:
To be turned in Feb 23:
3.2/18,24,28,36,40,48
3.3/24,30,32,36
3.4/6,8,26,28,34,42,44,46,50,56,58		 For additional practice:
3.2/1-3,6,17,19,37,41,49
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 - keyed to 4th Edition (T/F added)
Thurs, Feb 23
(Class #5)

Matrix of a linear transformation relative to an alternate basis, applications to linear transformations defined geometrically.

Intro. to 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.
HW #5: Read sections 3.4 and 4.1-4.3, and do problems:
To be turned in Mar 1:
3.4/60,62,70,72,74
4.1/20,26,30
4.2/6,25,52,53
4.3/22,27,28,47		 For additional practice:
3.4/59,69,71,73
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 - keyed to 4th Edition (T/F added)
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 1
(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 for additional details).

Midterm Exam #1
HW #6: Read sections 5.1, 5.2, and the first page of 5.4, and do problems:
To be turned in Mar 8:
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 - keyed to 4th Edition
Thurs, Mar 8
(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.

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 22:
5.2/8,14,22,28,34,44,45
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
5.3/45,47
5.4/15,17,18,24,37
Chapter 5 T/F Problems (corrected)
HW #7 problems - keyed to 4th Edition (T/F corrected)
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.
Thurs, Mar 15 No Class - Harvard Spring Break
Thurs, Mar 22
(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).
HW #8: Read sections 6.1 to 6.3 and do problems:
To be turned in Mar 29:
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
6.3/3,19,20,23,48,49
Chapter 6 T/F problems
HW #8 problems - keyed to 4th Edition (T/F corrected)
Thurs, Mar 29
(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.
HW #9: Read sections 7.1-7.4 and do problems:
To be turned in Apr 5:
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 - keyed to 4th Edition
Thurs, Apr 5
(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.
HW #10: Read sections 7.4-7.5 and do problems:
To be turned in Apr 12:
7.2/40,41
7.3/32,33,34,36,44
7.4/12,18,32,36,48,50
7.5/20,24,28,30,32		 For extra practice:
7.3/27,35,47
7.4/4-6,11,19,47,49
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 - keyed to 4th Edition (7.3/44 and T/F corrected)
Thurs, Apr 12
(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.
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 19:
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 - keyed to 4th Edition
Thurs, Apr 19
(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.

Midterm Exam 2
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 26:
8.1/5,19,20,29,36
8.2/4,6,8,9,11,16,18,22		 For extra practice and enlightenment:
8.2/1,2,3,15,19,21
Chapter 8 T/F problems
Problems #1 and #2 from supplement
HW #12 problems - keyed to 4th Edition
  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.
Thurs, Apr 26
(Class #13)

Systems of linear differential equations and their solutions - distinct real eigenvalue case, complex eigenvalue case.

Supplement on 1st order linear systems of differential equations and the use of evolution matrices
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 3:
9.1/24,(26,32),(28,34),(29,35),31,42,52  
(paired problems best done together)
9.2/6,7,12,31,36		 For extra practice and enlightenment:
9.1/4,5,13,21,22,23,43,49,54,55
9.2/22-26,34,39
HW #13 problems - keyed to 4th Edition
Thurs, May 3
(Class #14)

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

 HW #14: Read sections 9.1 and 9.2 and the supplement on nonlinear systems and linearization.
Thurs, May 10 FINAL EXAM (no alternate exam dates will be permitted)

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