Fall 2007 - Math 15a topics and assignments
[Assignments in grey and will be updated as the course proceeds.]
[last updated December 23, 2007]

Section 1 Date
(Section 2)		 Topics (dates and topics will shift as the course proceeds) Homework assignment 
The dates and content of these assignments are subject to change.
Thurs, Aug 30
Class #1
(Wed, Sept 5)		 Chapter 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 to 1.3 of the Bretscher text and do the following problems:

Due Monday, Sept 10 (Sept 12 - sect. 2):
1.1/11,16,17,25,29
1.2/9,10,16,30,34,36,42,60 
1.3/2,3,4,24,26,47,48		 For additional practice:
1.1/1,3,7,15   1.2/5,11,20-22   1.3/22,23
Economics problems: 1.1/20,21 and 1.2/37,38,39
HW #1 problems (PDF)

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.
Wed, Sept 5
Class #2
(Mon, Sept 10)		 Linear systems, continued - row reduction methods, 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.
Thurs, Sept 6
Class #3		 Product of matrix and vector, matrix form of a linear system.
Chapter 2: Linear transformation defined by a matrix; linearity property

Supplement on the dot product and orthogonal projection 
(for those who have not yet taken multivariable calculus)
Mon, Sept 10
Class #4
(Wed, Sept 12)		 Geometric meaning of linearity; elementary basis vectors; meaning of the columns of a matrix.
HW #2: Read sections 2.1 to 2.4 and do the following problems:
Problems due Mon, Sept 24(an extra day or two is OK):
2.1/8,24-30,43,44
2.2/4,5,6,7,19-23,34
2.3/2,4,6
2.4/15,16-25,34,44,46,48,49		 For additional practice:
2.1/5,9 and 2.2/10,11,27,28
2.3/40-43,53,54
2.4/1-4,7,10-12,14,28,29,31-33,36,63
Economics: 2.3/49,50;
Chap 2 True/False questions

HW #2 problems (PDF) - only problems on pgs 1-3 are to be turned in
Wed, Sept 12
Class #5
(Mon, Sept 17)		 Identity matrix, dilation (scaling) matrix. Constructing matrices of common linear transformations in geometry - rotations, dilations.
Mon, Sept 17
Class #6
(Wed, Sept 19)		 Algebraic and geometric properties of the dot product. Matrices for projections and reflections. Invertible linear transformations and algorithm for finding the inverse of a matrix (when it exists).
Wed, Sept 19
Class #7		 Composition of linear functions and matrix products. Matrix algebra.
Thurs, Sept 20
Class #8
(Mon, Sept 24)		 Chapter 3: Subspaces of Rn, span of a collection of vectors; kernel and image of a linear transformation.
HW #3a: Read sections 3.1 and 3.2 and do the following problems:
Due Wed, Oct 3 (or before Friday):
3.1/8,12,20,32,34,39,44
3.2/18,24,28,36,40,48
3.3/24		 For additional practice:
3.1/1,2,5,6,19,23-25,31
3.2/1-3,6,17,19,37,41,49
3.3/23

HW #3a problems (PDF)
Note: Problems 3.2/6,46 had been listed here but not on the PDF, so they have been omitted from the assignment.
Mon, Sept 24
Class #9		 Finding the kernel and image of a linear transformation.
Wed, Sept 26
Class #10
(sect. 1 only)		 Linear dependence and linear independence. Basis and dimension of a subspace; Rank-Nullity Theorem.
Mon, Oct 1
Class #11
(Mon, Oct 1)		 Coordinates of a vector relative to a basis; matrix of a linear transformation relative to an alternate basis.
HW #3b: Read sections 3.3 and 3.4 and do the following problems:
Due Thurs, Oct 11:
3.3/30,32,36
3.4/6,8,18,26,28,42,44,46,50,56,74		 For additional practice:
3.3/27,29,40,41,42,61
3.4/5,7,17,27,45,55
Chapter 3 True/False questions
HW #3b problems (PDF)
Wed, Oct 3
Class #12
(Wed, Oct 3)		 Interpretation of the columns of a matrix relative to a given basis; similarity of matrices. Matrix relative to a basis, examples and applications.
Practice Exam #1 
Mon, Oct 8
Class #13
(Mon, Oct 8)

Midterm Exam #1 in class covering Chapter 1-3 of the text.
Tues, Oct 9
Class #14
(sect. 1 only)		 Final details of coordinates and matrices relative to alternate bases.
Wed, Oct 10
Class #15
(Wed, Oct 10)		 Homework questions.
Thurs, Oct 11
Class #16		 Chapter 4: Introduction to general linear spaces, e.g. function spaces and families of matrices.
HW #4: Read sections 4.1-4.3 and do the following problems:
Due Mon, Oct 22:
4.1/20,26,30
4.2/6,25,52,53,66
4.3/22,27,28,47		 For additional practice:
4.1/1-3,9,10,11,25,29
4.2/2,4,67,81
4.3/1,13,14,44
Chapter 4 True/False questions
HW #4 problems (PDF)
An error in the hint in problem 4.2/6 has been corrected.
Mon, Oct 15
Class #17
(Mon, Oct 15)		 Linear transformations and isomorphisms, coordinates relative to a basis for a linear space or subspace. Image, kernel, rank, nullity of general linear transformations.
Wed, Oct 17
Class #18
(Wed, Oct 17)		 Last details of general linear spaces: isomorphisms, examples.
Thurs, Oct 18
Class #19		 Homework questions and linear spaces, continued.
Mon, Oct 22
Class #20
(Mon, Oct 22)		 Chapter 5: Dot product, orthogonality,  orthonormal bases and their advantages. 
HW #5a: Read sections 5.1-5.2 and do the following problems:
Due Thurs, Nov 1:
5.1/12,15,16,17,26,28
5.2/8,14,22,28,34,38		 For additional practice:
5.1/3,5,18,23,29
5.2/6,20,33,40,41,44,45

HW #5a problems (PDF)

Wed, Oct 24
Class #21
(Wed, Oct 24)		 Finding coordinates of a vector relative to an orthonormal basis; orthogonal projection; matrices for orthogonal projection and reflection using an ON basis.
Thurs, Oct 25
Class #22		 Gram-Schmidt process and QR factorization; orthogonal transformations; orthogonal matrices. Facts about transposes.
Mon, Oct 29
Class #23
(Mon, Oct 29)		 Method of Least Squares for approximate solutions to a linear system, regression lines, and data fitting; alternate method for finding matrix for orthogonal projection.
HW #5b: Read sections 5.3-5.4 and the Supplement on Least Squares. Do problems:
Due Thurs, Nov 8:
5.3/5-11,40,42,44,46,47,68
5.4/2,4,5,16,20,26,32,38		 For additional practice:
5.3/31,32,37,45,48
5.4/1,6,7,15,17,18,22,24,31,37,41,42
Chapter 5 T/F questions

HW #5b problems (PDF)

Wed, Oct 31
Class #24
(Wed, Oct 31)		 Homework questions; Method of Least Squares, continued; alternate method for finding matrix for orthogonal projection.
Thurs, Nov 1
Class #25

Regression lines and data fitting using the Method of Least Squares.
HW #6: Read chapter 6 and do the following problems: 
Due Thurs, Nov 15:
6.1/18,26,30,34,44
6.2/6,18,26
6.3/7,13,14,18		 For additional practice:
6.1/9,16,17,43
6.2/5,9,17,25,34,40,41,43
6.3/3,19,20,23,48,49
Chapter 6 T/F problems

HW #6 problems (PDF)

Mon, Nov 5
Class #26
(Mon, Nov 5)		 Chapter 6: Determinant of a square matrix, permutations, Laplace expansion, multilinearity, effect of row operations on the value of the determinant.
Wed, Nov 7
Class #27
(Wed, Nov 7)		 Invertibility of a matrix A 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. 
Thurs, Nov 8
Class #28		 Homework questions.
Practice problems for Exam #2 
Mon, Nov 12
Class #29
(Mon, Nov 12)		 Midterm Exam #2 covering Chapters 3 to 5.4 of the text plus basic properties and applications of determinants.
Wed, Nov 14
Class #30
(Wed, Nov 14)		 Chapter 7: 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. 
HW #7a: Read sections 7.1-7.4 and do problems:
To be turned in Wed, Nov 28:
7.1/33,36,50
7.2/22,23,28
7.3/2,4,8,14,36,44
7.4/12,18,32,48,50		 For additional practice:
7.1/1-6,15-21,39,53
7.2/5,7,15,24,25,26,27
7.3/11,16,21,27,32,33,35,47
7.4/4,5,6,11,19,47,49

HW #7a problems (PDF)

Thurs, Nov 15
Class #31

Discrete linear dynamical systems, powers of a matrix, and Markov matrix example. Phase portraits. 

Mon, Nov 19
Class #32
(Mon, Nov 19)		 Eigenvalues of a linear transformation; obstructions to diagonalizability. Similar matrices have the same characteristic polynomials and the same eigenvalues with the same algebraic and geometric multiplicities.
Wed, Nov 21
Class #33
(Wed, Nov 21)		 Trace and determinant in terms of eigenvalues; characteristic polynomial and eigenvalues of a linear transformation. 
HW #7b: Read sections 7.5 and 7.6 and the supplement on repeated eigenvalues, complex eigenvalues, and Jordan canonical form for a matrix.

To be turned in by Wed, Dec 5:
7.5/20,22,24,28,30,32
7.6/10,12,24,38

 For additional practice:
7.5/21,23,25,27,29,36
7.6/3,4,17,37
Chapter 7 T/F problems

HW #7b problems (PDF)

Mon, Nov 26
Class #34
(Mon, Nov 26)		 Dealing with repeated eigenvalues where GM < AM. Introduction to complex eigenvalues. Review of complex numbers; complex eigenvalues and invariant (rotation-dilation) subspaces.
Wed, Nov 28
Class #35
(Wed, Nov 28)		 Last details of complex eigenvalues. 
Thurs, Nov 29
Class #36		 Stability of a discrete linear dynamical system.
HW #8: Read sections 8.1 and 8.2.

Do, but don't turn in:
8.1/6,10,12,16,24,36
8.2/4,6,8,16,18,22

 For additional practice:
8.1/3,5,15,19,29
8.2/1,2,3,9,11,15,19,21
Chapter 8 T/F problems
HW #8 problems (PDF)

Practice Final Exam problems 
Mon, Dec 3
Class #37
(Mon, Dec 3)

Classes did not meet due to Clinton.
Wed, Dec 5
Class #38
(Wed, Dec 5)

Chapter 8: Spectral Theorem and orthogonal diagonalizability of symmetric matrices. Quadratic forms, positive definiteness, principal axes, and applications to ellipses, hyperbolas, and quadratic surfaces (ellipsoids, hyperboloids of one or two sheets).
Thurs, Dec 6
Class #39		 Wrapping up...
Fri, Dec 14 Final Exam: 9:15am to 12:15pm in Pollack Room 1.  

Note: There are 39 scheduled 50 minute classes for Section 1, and 26 scheduled 70 minute classes for section 2. The Calendar is keyed to Section 1, and the dates on which section 2 will cover the listed topics is approximate.

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