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 1.14 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 
Subvector
spaces Definition of a subvector 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
2.12. Read Section 3 of the following .pdf 

Friday January 26 
Dimension
and bases Definition of dimension, linear dependence, and bases. 
Apostol Sections
1.79 Read Section 4 of the following .pdf 
HW 2 Apostol Section 1.5: 14, 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 1.1112 

Wednesday
January 31 
Inner products
continued Construction of orthogonal sets, the GraemeSchmidt process, and orthogonal complements. 
Apostol Sections 1.1415 

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 NullityRank Theorem. 
Apostol Sections 2.13  
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 2.67  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 2.910  
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, 2224, 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 2.1315  
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 2.1819 
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
3.13. 

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 3.78 

Wednesday
March 14 
Determinants continued Determinants and independence of vectors. Determinants of blockdiagonal matrices. 
Apostol Section 3.910 

Friday March 16  Determinants continued Expansion formulas for determinants, minors, cofactors, determinant of a transpose, the cofactor matrix, and Cramer's rule. 
Apostol Sections 3.1216 
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 2630  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 4.23  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 CayleyHamilton Theorem. 
Apostol Section 4.5 

Wednesday April 11 
Eigenvalues
and eigenvectors continued Calculation of eigenvalues and eigenvectors, trace of a matrix. 
Apostol Sections 4.67  
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 5.12, 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 