Homework and Syllabus
Linear Algebra
MATH 2135 Fall 2020
Homework must be submitted via Canvas, and
is due at the start of class, with your name
and homework number on it to receive credit.
You will be graded on the clarity of your exposition.
Messy, disorganized, or poorly written assignments will not
receive credit.
Starting with Homework 3,
all assignments must be written in LaTeX.
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.
An asterix * indicates that a homework assignment has not been finalized.
| Date | Topics | Reading | Homework |
| L1 Monday
August 24 |
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. Reduced Row Echelon Form of a matrix. Introduction to LaTeX. |
You can review some of the standard
mathematical language in: Richard Hammack, The Book of Proof, Creative Commons, 2nd Edition, 2013. Here are some slides on the Reduced Row Echelon Form of a matrix. 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. You may also want to take a look at The Not So Short Introduction to LaTeX 2e |
|
| L2 Wednesday
August 26 |
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. |
|
| L3 Friday
August 28 |
Sub-vector spaces Definition of a sub-vector space. |
Apostol Section 1.6 Read Section 2 of the following .pdf |
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. 5 points extra credit for homework typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
| L4 Monday August 31 |
Linear
maps Definition of a linear map, kernel, and image. Linear map associated to a matrix. |
Apostol Sections
2.1-2. Read Section 3 of the following .pdf |
|
| L5 Wednesday September 2 | Dimension
and bases Definition of dimension, linear dependence, and bases. Connection to Reduced Row Echelon Form of a matrix. |
Apostol Sections
1.7-9 Read Section 4 of the following .pdf |
|
| L6 Friday September 4 |
Inner
products Definition of an inner product, Euclidean spaces, Norms, and orthogonality. |
Apostol Sections
1.11-12 |
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. 2 points extra credit for homework typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
| Monday September 7 | NO CLASS |
LABOR DAY | |
| L7 Wednesday
September 9 |
Inner products
continued Construction of orthogonal sets, the Gram--Schmidt process, and orthogonal complements. |
Apostol Sections 1.14-15 |
|
| L8 Friday
September 11 |
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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
| L9 Monday September 14 |
Linear
maps continued Definition of a linear map, kernel, and image. The Rank--Nullity Theorem. |
Apostol Sections 2.1-3 | |
| L10 Wednesday September 16 | Linear
maps continued Linear maps as a vector space |
Apostol Section 2.5 | |
| L11 Friday September 18 |
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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
| L12 Monday September 21 |
Review | |
|
| L13 Wednesday September 23 |
Review | Sample Midterm I .pdf Solutions .pdf |
|
| Friday September 25 |
MIDTERM I | MIDTERM I | MIDTERM I |
| L14 Monday September 28 |
Review
of exam |
||
| L15 Wednesday September 30 |
Linear
maps continued Linear maps with prescribed values, matrix representation of a linear map. |
Apostol Sections 2.9-10 | |
| L16 Friday October 2 |
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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
| L17
Monday October 5 |
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 | |
| L18
Wednesday October 7 |
Computational topics Systems of linear equations. |
Apostol Section 2.17 |
|
| L19
Friday October 9 |
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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
| L20 Monday October 12 |
Determinants Introduction, and axioms. |
Apostol Sections
3.1-3. |
|
| L21 Wednesday October 14 |
Determinants
continued Computation of determinants. |
Apostol Section 3.4 |
|
| L22 Friday October 16 |
Determinants
continued The uniqueness theorem. |
Apostol Section 3.5 |
HW 7 Apostol Section 2.20: 2, 4, 11, 12, 16. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
| L23
Monday October 19 |
Determinants continued Product formula, determinant of an inverse. |
Apostol Section 3.7-8 |
|
| L24
Wednesday October 21 |
Determinants continued Determinants and independence of vectors. Determinants of block-diagonal matrices. |
Apostol Section 3.9-10 |
|
| L25 Friday October 23 | 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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas Recommended, but not to be turned in: Section 3.6: 2, 3, 9. Section 3.11: 6. |
| L26 Monday October 26 |
Review |
||
| L27 Wednesday October 28 |
Review |
Sample Midterm II .pdf Solutions .pdf |
|
| Friday October 30 |
MIDTERM
II |
MIDTERM II | MIDTERM II |
| L28
Monday November 2 |
Review of exam |
||
| L29
Wednesday November 4 |
Eigenvalues and
eigenvectors Introduction, linear transformations with diagonal matrix representations. |
Apostol Section 4.1 |
|
| L30
Friday November 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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas Recommended, but not to be turned in: Section 3.17: 6, 8. |
| L31 Monday November 9 |
Eigenvalues
and eigenvectors continued Characteristic polynomials, the Cayley--Hamilton Theorem. |
Apostol Section 4.5 |
|
| L32 Wednesday November 11 |
Eigenvalues
and eigenvectors continued Calculation of eigenvalues and eigenvectors, trace of a matrix. |
Apostol Sections 4.6-7 | |
| L33 Friday November 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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas Recommended, but not to be turned in: Section 4.4: 1, 11, 12. Section 4.8: 1, 12, 14. |
| L34
Monday November 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. |
|
| L35
Wednesday November 18 |
Applications of
linear algebra: Quadratic forms, and maximizing their value subject to constraints. |
||
| L36
Friday November 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. Homework must be typed in LaTeX .tex file and .pdf file must be uploaded to Canvas |
|
| L37 Monday November 23 |
Applications
of linear algebra: Singular value decomposition, and applications. |
||
| L38 Wednesday November 25 |
Applications
of linear algebra: Linear regression, and applications. |
||
| Friday November 27 |
NO
CLASS |
THANKSGIVING BREAK | |
| L39 Monday November 30 |
Applications of linear
algebra: Markov chains, and applications to the Google page rank algorithm. |
||
| L40
Wednesday December 2 |
Review | ||
| L41
Friday December 4 |
Review | 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. |
|
| L42 Monday December 7 | Review | Sample Final .pdf Solutions .pdf |
|
| Saturday
December 12 |
FINAL
EXAM Saturday December 12, 1:30 PM -- 4:00 PM |
FINAL EXAM |
FINAL EXAM |
We will be using LaTeX for
typing homework. If you have a mac,
one possible easy way to get started is with texshop.
You will want to download the MacTex
package. 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. 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.overleaf.com
(formerly sharelatex) for a cloud version.