Sebastian Casalaina

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


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


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


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

Wednesday February 14
Review Sample Midterm I .pdf

Solutions .pdf

Friday February 16
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


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


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
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


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


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

Wednesday March 21
Sample Midterm II .pdf

Solutions .pdf

Friday March 23
March 26--30 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


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


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


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):


Section 5.5: 1, 3, 7, 9.
Section 5.11: 1, 2, 3, 14.
Monday April 30

Wednesday May 2
Review Sample Final .pdf

Solutions .pdf

Friday May 4 NO CLASS
Monday May 7
FINAL EXAM 1:30 PM - 4:00 PM ECCR 105 (Lecture Room) 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:  You may also want to try (formerly or for a cloud version.