Sebastian Casalaina

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.