Homework and Syllabus
Linear Algebra
MATH 2135 Spring 2018
Homework is due in class and must be stapled
with your name and homework number on it to
receive credit. You will be graded on the clarity of
your exposition.
Please read the suggested texts before class, and
then after class make sure to attempt the homework for the
sections we covered that day.
You may find it useful to use a computer algebra system to
check your matrix computations. The program Mathematica,
for instance, is available free to students via
the University of Colorado.
You may find the Mathematics
Academic Resource Center ("MARC" MATH 175) to be
useful as a meeting point for discussing homework.
An asterix * indicates that a homework assignment has not been finalized.
| Date | Topics | Reading | Homework |
| Wednesday
January 17 |
Introduction to the
course, and review of mathematical notation Sets, subsets, unions, intersections, products, equivalence relations, and maps. Matrix multiplication, and determinant and inverse formulas for small matrices. Introduction to LaTeX. |
You can review some of the standard
mathematical language in: Richard Hammack, The Book of Proof, Creative Commons, 2nd Edition, 2013. Please see the references for LaTeX at the bottom of this webpage. If possible, bring a laptop with LaTeX installed. Here are some sample files we will use: .tex, .bib, .pdf. |
|
| Friday
January 19 |
Definition of a vector
space Vector spaces, examples, and elementary consequences of the axioms. |
Tom M. Apostol, Calculus,
Volume 2 (2nd
Edition), Wiley, 1969, Sections 1.1-4 Read Section 1 of the following .pdf, which gives a brief overview of vector spaces and linear maps. |
HW 1 Hammack Chapter 12 Section 12.1: 4,6. Section 12.2: 5,10,16. Section 12.4: 2,6,10. Section 12.5: 2,8. |
| Monday January 22 |
Sub-vector
spaces Definition of a sub-vector space. |
Apostol Section 1.6 Read Section 2 of the following .pdf |
|
| Wednesday January 24 |
Linear
maps Definition of a linear map, kernel, and image. |
Apostol Sections
2.1-2. Read Section 3 of the following .pdf |
|
| Friday January 26 |
Dimension
and bases Definition of dimension, linear dependence, and bases. |
Apostol Sections
1.7-9 Read Section 4 of the following .pdf |
HW 2 Apostol Section 1.5: 1-4, 23, 24, 31 (a), (b). Section 2.4: 2, 4, 6, 12, 13, 16, 18, 28, 29. |
| Monday
January 29 |
Inner products Definition of an inner product, Euclidean spaces, Norms, and orthogonality. |
Apostol Sections 1.11-12 |
|
| Wednesday
January 31 |
Inner products
continued Construction of orthogonal sets, the Graeme--Schmidt process, and orthogonal complements. |
Apostol Sections 1.14-15 |
|
| Friday
February 2 |
Inner products continued Projections, and best approximations. |
Apostol Section 1.16 |
HW 3 Apostol Section 1.10: 1, 4, 10, 11, 17, 18, 24. Section 1.13: 1, 4, 8, 13, 15, 16. 5 points extra credit for homework typed in LaTeX |
| Monday February 5 |
Linear
maps continued Definition of a linear map, kernel, and image. The Nullity--Rank Theorem. |
Apostol Sections 2.1-3 | |
| Wednesday February 7 |
Linear
maps continued Linear maps as a vector space |
Apostol Section 2.5 | |
| Friday February 9 |
Linear
maps continued Injectivity, and inverses. |
Apostol Section 2.6-7 | HW 4 Apostol Section 1.17: 2, 3, 6, 8. Section 2.4: 1, 2, 6, 12, 24, 26. 2 points extra credit for homework typed in LaTeX |
| Monday February 12 |
Review | |
|
| Wednesday February 14 |
Review | Sample Midterm I .pdf Solutions .pdf |
|
| Friday February 16 |
MIDTERM I | MIDTERM I | MIDTERM I |
| Monday February 19 |
Review
of exam |
||
| Wednesday February 21 |
Linear
maps continued Linear maps with prescribed values, matrix representation of a linear map. |
Apostol Sections 2.9-10 | |
| Friday February 23 |
Linear
maps continued Construction of a matrix representation in diagonal form, review. |
Apostol Section 2.11 | HW 5 Apostol Section 2.8: 1, 3, 4, 8, 22-24, 27, 30. 2 points extra credit for homework typed in LaTeX |
| Monday
February 26 |
Linear maps continued Isomorphism between the vector space of linear maps and the vector space of matrices. Identification of composition with matrix multiplication. |
Apostol Sections 2.13-15 | |
| Wednesday
February 28 |
Computational topics Systems of linear equations. |
Apostol Section 2.17 |
|
| Friday
March 2 |
Computational topics
continued Computation techniques, inverses of square matrices. |
Apostol Sections 2.18-19 |
HW 6 Apostol Section 2.12: 1(c), 2, 8, 11, 17. Section 2.16: 1, 2, 3(a), 4(a), 5, 6, 7, 12. 2 points extra credit for homework typed in LaTeX |
| Monday March 5 |
Determinants Introduction, and axioms. |
Apostol Sections
3.1-3. |
|
| Wednesday March 7 |
Determinants
continued Computation of determinants. |
Apostol Section 3.4 |
|
| Friday March 9 |
Determinants
continued The uniqueness theorem. |
Apostol Section 3.5 |
HW 7 Apostol Section 2.20: 2, 4, 11, 12, 16. 2 points extra credit for homework typed in LaTeX |
| Monday
March 12 |
Determinants continued Product formula, determinant of an inverse. |
Apostol Section 3.7-8 |
|
| Wednesday
March 14 |
Determinants continued Determinants and independence of vectors. Determinants of block-diagonal matrices. |
Apostol Section 3.9-10 |
|
| Friday March 16 | Determinants continued Expansion formulas for determinants, minors, cofactors, determinant of a transpose, the cofactor matrix, and Cramer's rule. |
Apostol Sections 3.12-16 |
HW 8 Apostol Section 3.6: 1, 3, 4(a), 5, 6. Section 3.11: 1, 2, 5. 2 points extra credit for homework typed in LaTeX Recommended, but not to be turned in: Section 3.6: 2, 3, 9. Section 3.11: 6. |
| Monday March 19 |
Review |
||
| Wednesday March 21 |
Review |
Sample Midterm II .pdf Solutions .pdf |
|
| Friday March 23 |
MIDTERM
II |
MIDTERM II | MIDTERM II |
| March 26--30 | SPRING BREAK |
SPRING BREAK |
SPRING BREAK |
| Monday
April 2 |
Review of exam |
||
| Wednesday
April 4 |
Eigenvalues and
eigenvectors Introduction, linear transformations with diagonal matrix representations. |
Apostol Section 4.1 |
|
| Friday
April 6 |
Eigenvalues and
eigenvectors continued Eigenvalues and eigenvectors defined, linear independence of eigenvectors with distinct eigenvalues. |
Apostol Section 4.2-3 | HW 9 Apostol Section 3.17: 1(a),(b), 2(a),(b), 3, 4, 5, 7. 2 points extra credit for homework typed in LaTeX Recommended, but not to be turned in: Section 3.17: 6, 8. |
| Monday April 9 |
Eigenvalues
and eigenvectors continued Characteristic polynomials, the Cayley--Hamilton Theorem. |
Apostol Section 4.5 |
|
| Wednesday April 11 |
Eigenvalues
and eigenvectors continued Calculation of eigenvalues and eigenvectors, trace of a matrix. |
Apostol Sections 4.6-7 | |
| Friday April 13 | Eigenvalues
and eigenvectors continued Matrices representing the same linear transformation, similar matrices, review. |
Apostol Section 4.9 | HW 10 Apostol Section 4.4: 2, 3, 5, 7, 9. Section 4.8: 3, 4, 5, 7. 2 points extra credit for homework typed in LaTeX Recommended, but not to be turned in: Section 4.4: 1, 11, 12. Section 4.8: 1, 12, 14. |
| Monday
April 16 |
Eigenvalues of
operators acting on Euclidean spaces Overview of Hermitian operators, and the spectral theorem. |
Apostol Sections 5.1-2, and Theorems
5.4, 5.7. |
|
| Wednesday
April 18 |
Applications of
linear algebra: Quadratic forms, and maximizing their value subject to constraints. |
||
| Friday
April 20 |
Applications of
linear algebra: Principal component analysis, with applications to statistics and image processing. |
HW 11 Apostol Section 4.10: 1, 2, 4, 7, 8. 2 points extra credit for homework typed in LaTeX |
|
| Monday April 23 |
Applications
of linear algebra: Singular value decomposition, and applications. |
||
| Wednesday April 25 |
Applications
of linear algebra: Linear regression, and applications. |
||
| Friday April 27 |
Applications
of linear algebra: Markov chains, and applications to the Google page rank algorithm. |
HW 12 Review for final exam Here are a few problems to look at for fun on Hermitian operators (not to be turned in): Apostol Section 5.5: 1, 3, 7, 9. Section 5.11: 1, 2, 3, 14. |
|
| Monday
April 30 |
Review | ||
| Wednesday
May 2 |
Review | Sample Final .pdf Solutions .pdf |
|
| Friday May 4 | NO CLASS |
NO CLASS |
NO CLASS |
| Monday
May 7 |
FINAL EXAM 1:30 PM - 4:00 PM ECCR 105 (Lecture Room) | FINAL EXAM |
FINAL EXAM |
I strongly encourage
everyone to use LaTeX for typing homework. If you have
a mac,
one possible easy way to get started is with texshop.
If you are using linux,
there are a number of other possible ways to go, using
emacs, ghostview, etc. If you are using windows,
you're on your own, but I'm sure there's something online.
Here is a sample homework file to use: (the .tex
file, the .bib
file, and the .pdf
file). This site can help you find LaTeX symbols
by drawing: http://detexify.kirelabs.org/classify.html.
You may also want to try https://cocalc.com
(formerly https://cloud.sagemath.com/)
or https://www.sharelatex.com/
for a cloud version.