Instructor: David A. Meyer
Office hours: AP&M 7256, WTh 1:00pm-2:00pm, or by appointment
Lecture: AP&M, room B412 (basement), MWF 9:00am-9:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu
Teaching assistant: Dan Schultheis
Office hours: AP&M 5132, M 2:00pm-4:00pm, Th 9:00am-11:00am, or by appointment
Section A01: WLH, room 2209, Tu 2:00pm-2:50pm
Section A02: WLH, room 2209, Tu 3:00pm-3:50pm
Email: dschulth "at" math "dot" ucsd "dot" edu
This course is the first in the Mathematics Department's Honors sequence. It covers basic linear algebra: vectors and matrices, solution of systems of linear equations, geometry of vector spaces, linear transformations, eigenvalues and eigenvectors.
The three courses in the Honors sequence, Math 31ABC, cover essentially the same material as do Math 20F, 20C, and 20E, respectively, but at a more sophisticated conceptual level. The Honors sequence emphasizes proofs, so students completing it will be exempt from taking Math 109 (Mathematical Reasoning). The prerequisite is AP calculus in high school, with a 5 on the BC exam, or permission from the instructor. A grade of B- or better is necessary to continue from one course to the next in the sequence. The Honors sequence is intended for mathematics majors and prospective mathematics majors (although others are very welcome) and gives a much better view of what upper-division mathematics is like than does the standard Math 20 sequence.
The textbook for the whole Honors sequence is J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Third Edition (Ithaca, NY: Matrix Editions 2007). Errata have been compiled. In this course we will cover at least Chapter 0, sections 0, 1, 3, 4, 7; Chapter 1, sections 0—4; Chapter 2, sections 0—7. There is a copy of the textbook on reserve at the Science and Engineering library. There are many other linear algebra textbooks, most of which cover approximately the same material, at different levels of rigor and with some differences in emphasis (see, e.g., Hoffman and Kunze [1] or Strang [2]). Also, the Math Department has a calculus lab in AP&M B402A which is staffed by TAs from the Math 10 and 20 calculus sequences; they may be able to help with your questions when neither David nor Dan is available, but some of the material we cover may be outside the range of questions for which they are prepared.
There will be weekly homework assignments, due in section on Tuesdays, or before then in the drop box on the sixth floor of AP&M. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. Homework scores will constitute 1/6 of the final grade.
There will be two midterms, approximately in the fourth and eighth weeks of the quarter. The final is scheduled for 8:00am Wednesday 9 December 2009. Scores on the two midterms and final will constitute 1/4, 1/4 and 1/3 of the final grade, respectively. There will be no makeup tests.
| 5 Nov 09 |
mathematics careers seminar AP&M B412, 4:00pm-5:00pm |
| 25 Sep 09 |
§1.0. Introduction overview of the course §0.3. Set theory notation Russell's paradox [3,4,5] §0.4. Functions definition one-to-one, onto HWK (due Tu 29 Sep 09). §0.3: 1 §0.4: 1, 2, 3, 4, 6 [solutions] |
| 28 Sep 09 |
inverse and pre-image §1.1. Points and vectors positions vs. increments example subtraction, addition and scalar multiplication subspaces of Rn |
| 30 Sep 09 |
examples the standard basis vectors §0.1. Summation notation vector fields example §1.2. Matrices definition addition and scalar multiplication multiplication HWK (due Tu 6 Oct 09). §1.1: 1, 4, 5, 6abch, 8 §1.2: 2, 3, 5, 8, 10, 15, 16 [solutions] |
| 2 Oct 09 |
multiplication definition non-commutativity associativity identity matrix matrix inverses |
| 5 Oct 09 |
matrix transpose symmetric, antisymmetric, triangular, diagonal matrices applications of matrix multiplication income mobility [6,7] counting paths in graphs |
| 7 Oct 09 |
mathematical induction Peano axioms [8] examples HWK (due Tu 13 Oct 09). §1.2: 17, 20, 21, * §1.3: 4, 8, 9, 12, 13, 18, 19, 20 [solutions] Extra Credit (due W 21 Oct 09). rounding with constraints |
| 9 Oct 09 |
§1.3. Linear transformations definition linear transformations are matrix multiplications geometrical meaning |
| 12 Oct 09 |
§0.4. Composition of functions definition matrix multiplication as composition of linear transformations associativity |
| 14 Oct 09 |
invertibility §1.4. Geometry of Rn length dot product HWK (due Tu 20 Oct 09). §0.4: 9, 10 §1.4: 3, 5, 7, 8, 10, 13, 16, 19, 24, 26 [solutions] |
| 16 Oct 09 |
Schwartz' inequality angles |
|
18 Oct 09 (Sunday) |
review session 1:00pm-3:00pm, HSS 1128A (ground level) (Dan's office hours on Th 22 Oct are cancelled) |
| 19 Oct 09 |
invariance of dot product under rotations of R2 geometrical meaning of determinant in 2 dimensions definition of determinant in 3 dimensions definition of cross product geometrical meaning of determinant in 3 dimensions review |
| 21 Oct 09 |
Midterm 1, covering §0.1, 0.3, 0.4, 1.0—1.4 |
| 23 Oct 09 |
Midterm 1 solutions HWK (due Tu 27 Oct 09). §2.1: 2, 3, 5, 8 [solutions] |
| 26 Oct 09 |
§2.1. Row reduction representing systems of linear equations as Ax = b row operations échelon form Extra Credit (due M 2 Nov 09). geometry of linear equations |
| 28 Oct 09 |
§2.2. Solving equations using row reduction no solutions unique solution infinite number of solutions HWK (due Tu 3 Nov 09). §2.2: 3, 5, 7, 11 §2.3: 2, 3, 5, 8, 12 [solutions] |
| 30 Oct 09 |
uniqueness of échelon form definition of pivotal column/variable §2.3. Matrix inverses used to solve equations A is invertible implies it row reduces to I definition of elementary matrices |
| 2 Nov 09 |
inverses of elementary matrices A row reduces to I implies it is invertible finding the inverse of a matrix §2.4. Linear combinations, span, and linear independence definition of linear combination definition of span examples |
| 4 Nov 09 |
definition of linear independence examples more than n vectors in Rn are linearly dependent fewer than n vectors in Rn cannot span HWK (due Tu 10 Nov 09). §2.4: 2, 3, 5, 7, 10, 11 (skip part c if you would have to do it by hand), 12 [solutions] |
| 6 Nov 09 |
definition of basis definition of orthogonal and orthonormal each subspace of Rn has a basis every basis of a subspace has the same cardinality, the dimension of the subspace Extra Credit (due M 16 Nov 09). orthogonal groups |
| 9 Nov 09 |
§2.5. Kernel and image definition of kernel and image of linear transformation T(x) = b has no more than 1 solution for all b iff kerT = {0} T(x) = b has at least 1 solution for all b iff imgT is the codomain of T a basis for imgT is given by the pivotal columns in [T] HWK (due Tu 17 Nov 09). §2.5: 1, 2, 3, 5, 6, 8, 9, 10, 15 18 [solutions] |
| 11 Nov 09 |
no lecture, Veterans Day holiday [9,10] |
| 13 Nov 09 |
finding a basis for kerT for any linear transformation T :Rn → Rm, dim kerT + dim imgT = n rankT = dim imgT, nullityT = dim kerT for any matrix A, rankA = rankAT polynomial interpolation define T :{polymomials of degree k} → Rk+1 by T(p) = [p(0),p(1),...,p(k)] T is a linear transformation |
|
15 Nov 09 (Sunday) |
review session 1:00pm-3:00pm, HSS 1128A (ground level) practice midterm |
| 16 Nov 09 |
kerT = {0}, so T -1 exists calculation of T -1 for k = 2 review |
| 18 Nov 09 |
Midterm 2, covering §2.0—2.5 |
|
20 Nov 09 Dan |
Midterm 2 solutions |
| 23 Nov 09 |
§2.6. Abstract vector spaces definition of real vector space definition of linear transformation, isomorphism example of a vector space, continuous functions on [0,1], not isomorphic to Rn HWK (due Tu 1 Dec 09). §2.6: 1, 2, 4, 5, 7, 8, 10, 11 |
| 25 Nov 09 |
definition of linear combination, span, linear independence, basis definition of the concrete-to-abstract function derivation of the change-of-basis function |
| 27 Nov 09 |
no lecture, Thanksgiving holiday |
| 30 Nov 09 |
§2.7. Eigenvalues and eigenvectors Fibonnaci numbers example raising a matrix to a power definition of eigenvalue and eigenvector a matrix is diagonalizable iff a matrix of its eigenvectors is invertible |
| 2 Dec 09 |
eigenvectors with distinct eigenvalues are linearly independent finding eigenvalues and eigenvectors of a matrix Suggested problems. §2.7: 1, 2, 3, 6 |
| 4 Dec 09 |
§2.7. Eigenvalues and eigenvectors |
| [1] | K. M. Hoffman and R. Kunze, Linear Algebra, Second edition (Englewood Cliffs, NJ: Prentice Hall 1971). |
| [2] | G. Strang, Introduction to Linear Algebra, Fourth edition (Wellesley, MA: Wellesley-Cambridge Press 2009). |
| [3] | B. Russell, "Letter to Frege" (1902), in J. van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Cambridge, MA: Harvard University Press 1967) 124-125. |
| [4] | A. D. Irvine, "Russell's paradox", in E. N. Zalta, Principal Editor, Stanford Encyclopedia of Philosophy (2009). |
| [5] | A. Doxiadis and C. H. Papadimitriou, art by A. Papadatos and A. Di Donna, Logicomix: An Epic Search for Truth (New York: Bloomsbury USA 2009). |
| [6] | T. Hertz, "Understanding mobility in America", Center for American Progress report (2006). |
| [7] | J. B. Isaacs, "Economic mobility of families across generations", Pew Charitable Trusts report (2007). |
| [8] | D. R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid (New York: Basic Books 1979). |
| [9] | M. Cleland, "The forever war of the mind", New York Times, 6 November 2009. |
| [10] | C. Alexander, "Back from the war, but not really home", New York Times, 7 November 2009. |