Elements of Linear Algebra
Elements of Linear Algebra, MAT313, 4 credits
Section ZI81 7:459:25 T Th
Prerequisite: MAT 176
Text: Kolman, Introductory Linear Algebra with Applications,
6th edition, Prentice Hall
Course Webpage: select Elements of Linear Algebra from
my webpage. 
Professor C. Sormani
Office Hours: 5:306:00pm TuTh
9:3010:00pm TuTh
Office: Gillet Hall 200B
Email: sormani@comet.lehman.cuny.edu
Phone: 7189607422
Webpage:
http://comet.lehman.cuny.edu/sormani 
Grading Policy: Exam I: 25%... Exam II: 25%... Project: 20%...
Classwork: 5%... Final: 25%...
Each lesson has a corresponding reading assignment and homework
problems. The problems will not be collected. However, it is
expected that you will do the homework and check your answers in the
text. You should ask questions in class and in my office
hours when you have difficulty. Problems with a * will be put on
the board by students and gone over at the beginning of class. .
There will be two exams given in class. The best way to prepare
for the exams is to review your class notes, the corresponding chapters
in the textbook and practice problems without consulting your notes.
Always start studying at least 4 days before the exam and bring
questions to the review class. The final exam will be similar
to the exams, emphasizing the material learned after the second exam but
covering the rest of the semester as well. Be sure to review your old exams
before taking the final. The
project will be a semester long assignment
with four components and due dates. The classwork will consist of
in class participation and homework problems which are written on
the board before class begins. Extra credit assignments will
be given from time to time.
Syllabus: Vector spaces, systems of linear equations, determinants,
linear transformations, and matrices.

Short Week 1: 1.1 Systems of Linear Equations
8/31:
Read 1.1/ex 1, 2, 7 and pp 2830. Do 1.1/*1,3,7,*9,*15,19,*23,*T.4 1.3/*T.11,*T13ab

Week 2: 1.21.3 Matrices
9/5: Read 1.2, do 1.1/5, 11; 1.2/1,
5abe*,7abc*, *T1, *T4, *T5a
9/7: Read 1.3 pages 1820, ex 9,
linear systems, and ex 13,
Do 1.2/5c, T5bc, *T6ab, 1.3/1a,
*5ab, *9, *15, 19, *27, T1, *T4, T7,

Week 3: 1.4 Properties of Matrices
9/12: Read theorems and defn in
1.4. Do 1.3/1b, 5c, 29, *T6 1.4/*2, 5, *15, T1,*T2,
9/14: read 1.4: prove Thm
1.2 a using summation notation, Example 9, Do: 1.3/T2, *T3a, *T10,
T13c, 1.4/*7, 9, *13a, *19, *T3, *T14
Read the intro to each of the projects
listed below and choose one before 9/19. You should read all the intros
and then go a little further in your chosen topic.

Week 4: 1.51.6 Solutions and Inverses
9/19: Read 1.5, Do 1.3/*T9a, T3b, 1.4/*T19,
1.5/1,3,7*,9a*,9b*, T9*
9/21: Read 1.6, 1.3/ T13 (with n=3 and m=4),
1.4/18, *T32, 1.5/*9c, 15a, 17a,
1.6/1, *3, *5b, *7a, 7b, *19, *25

Week 5: 2.12.2 Determinants
9/26: Read 2.1, Do 1.6/5ac, *20, 2.1/1,5,*8,*15a, *T3
9/28: Read 2.2 up to page 112, Do 1.3/27*, 1.4/*T21, T23, *T26,
1.6/7c, 2.1/3, 19a, *T13 2.2/*3a,*9

Week 6: 2.3, Review and Exam I on Thursday.
There will be five problems: 1) solving a linear system, 2) matrix
multiplication with an application (like 1.3/example 9) 3) a proof
similar to the ones in 1.4 (know the theorems and defns), 4) finding
an inverse, 5) taking determinants and using this information
10/3: Review Homework: 1.3/28,29,30 1.5/T15, T16a, 1.6/T9,
1.6/9bc, 1.5/9abc, 2.2/3b, 11c
10/5: Exam I: On Chapters 12
Phase 1 of the project is due on 10/12.

Short Week 7: 3.13.2: Vectors
10/10: no class monday schedule
10/12:
Read: 127130, 132139, 141149.
Do: 3.1/2a, *5a(sketch), 9a, *19a, *21a,
3.2/1a, 7a*, 11a, 15*, 21a, 27a, T7
Phase 2 of the project is due on 10/17.

Week 8: 3.33.4: Linear Transformations and Computer
Graphics
10/17:
Read 3.3,
Do: 3.1/5c(sketch), 9b, 19b, 21d
3.2/T6, 3.3/*T2, *1ac, *17, *19, *23
10/19:
Read 3.4 and 3.6 pages 187191,
Do: 1.6/9a, 3.2/7b, 23, 27b, *T8,
3.3/*3abc, 25 3.4/*1c, *3, *4,
Extra Credit: Find the standard matrix for a transformation
that rotates a 3 vector around the z axis by 90 degrees and call it A, find the
standard matrix for a 3 vector that rotates a 3 vector around
the x axis by 90 degrees and call it B, Is AB=BA? Check your answer with a
picture and by taking the products.

Week 9: 4.1, 4.2, 6.1: Vector Spaces
10/24: Read 4.1, Do 3.4/1d, 4.1/*2, *3, 5, 8, *9, *11, 17, *T1, *T4
10/26: Read 4.2 and 6.1,
Do: 4.2/ 1, *7, *15, *16, *19, *21a, *T3, 6.1/ 1, 3a, *9, 13

Week 10: 4.34.4 Basis and Dimension
10/31: Read 4.3 up to Example 12, Do: 4.2/*20, *T5, 6.1/ *T3,
4.3/*1, *5, *T1, T4, *T7
11/2: Read 4.4, Do: 1.6/ 7c, 4.2/ *T9, 6.1/ *T5,
4.3/3, 7, *T13, 4.4/1ac, 3a, *7ab, *11, *17a,
Practice old homework for Exam II.

Week 11: Review and
Exam II on Thursday.
There will be five problems. 1) Adding vectors, finding the
lengths of vectors and angles between them, sketching. 2) Graphics,
sketching the images of vectors under linear transformations, finding
the standard matrix. 3) Verifying that a transformation is a linear
transformation. 4) Verifying that a subspace of a vector space we've
studied is a subspace (checking closure). 5) Verifying that a set of
vectors is linearly independant, that they span a given set, and that they
are a basis.
11/7: Vote even if you will be late for class!
11/9: Exam II focusing on 3.13.3, 6.1 and
4.14.4, will end promptly at 9:25 pm.
Phase 3 of the project is due on 11/16.

Week 12: 4.54.8
Rank and Orthogonality
11/14: Read 4.5 (skip Ex. 2 and 3)
and 4.6, Do 4.4/17b, 4.5/1, *3, 5, 7, *9, 4.6/1, *3, *9, *11
11/16: Read 4.8, Do 1.6/*5b, 4.3/T9,
4.4/31, 4.6/13, 15, 4.8/1a*, 3*, 5*, 19*

Short Week 13: 4.9 Orthogonal Matrices
11/21: Read 4.9 and 3.3 Exercise 1,
Do 4.6/*17, *19, 4.8/ 7, *11, *T5, 4.9/ *2(graph), *5(row and null
spaces only), *7abc, T5, Extra Credit: 4.9/T6.
11/23: no class, Thanksgiving

Week 14: 6.2 Kernel and Range
11/28: Review 6.1 and read 6.2, 6.1/3a*, 9*, T4*,
6.2/1, 3*, 7*, 11*, 19, T3*, T4*, T5*,
11/30: Read 5.1 pages 291299 and Read 4.5 Exercises 2 and 3,
1.6/7c, 4.4/ 29, 6.2/9, 5.1/ 1, 5*, 7*, 9*, 11*, T4*(3x3 case), T6*
Phase 4 of the project is due 12/7.

Week 15:
5.15.2 Eigenvalues and Eigenvectors
12/5: Read 5.1 pages 299309,
5.1/13*, 15*, 17*, 23*,
12/7: Read 5.2,
and pages 356358, 5.1/19*, 21*, 25*, T12*, T15*, 5.2/1, 5*, 7*,9*, T1, T5*
Extra Credits due on Tuesday Dec 19:
 I. Read 8.6, Do 8.6/1, 2*, 3, 4*, T1*
 II. Read 8.8, Do 8.8/1, 2*, 5, 6*, 23, 24*

Short Week 16: Catch up and Review
12/12: Review for the final: You may bring one 8.5x11in sheet
of paper filled with notes on one side to the final.
Please bring a xeroxed copy of the sheet to
hand in with the exam. Study the following topics and create one page
of notes for each, then compile these notes into a single sheet.

Solving Homogeneous systems, Reduced Row Echelon, finding independant
and dependant variables, writing a soln as a linear combination of
vectors, relationship to nonhomogeneous systems, Read 4.5, Example 1
and see page 242, This will be tested as part of an eigenvalue problem,
kernal problem and/or null space problem. Study this first!
 Using and Finding
Inverses of Matrices to solve linear systems of equations:
Read 1.6, 5.2, Do 1.6/25, p87/17, p86/13, 7

Linear Transformations, Kernals and Ranges, Standard Matrices,
Null Spaces and Column Spaces: Read 4.6, 6.1 and 6.2,
Do 6.2/Examples 11 and 12.

Vector Spaces,
Basis, span, linear independence, orthonormal basis, Gram Schmidt:
Read 4.3, 4.4, 4.8, Do 4.4/Examples 2 and 4, use procedure on p273,
Do 4.8/ Examples 4 and 5.

Eigenvalues, Eigenvectors, Diagonalization, using determinants
and solving homogeneous systems to find eigenspaces:
Read 2.1, 4.5, 5.15.2, Do 2.1/ Examples 5, 6 and 7,
4.5/Example 2, 5.1/Example 6, procedure page 305, 5.1/1, 7, 13, 5.2/Example 5.

Proofs and Theory: Summation Notation, Defn of a Linear Transformation,
Defn Vector Space, Defn of Matrix Multiplication with sum notation,
Read p2829, 1.3, 1.4, 4.1, 6.1,
Do 1.3/T4, 1.4/T7, 4.1/Examples 3 and 9, 5.2/T1, 6.1/T4.

Extra Office hour 89pm Tuesday Dec 19
12/19: Bring questions.

Final: Thursday Dec 21 810pm G205,
You may bring one 8.5x11in sheet
of paper filled with notes on one side to the final.
Please bring a xeroxed copy of the sheet to
hand in with the exam. No Calculators. There will be five multipart
problems. See review topics above.
Projects:

Marcov Chains (only if you have taken probability)
Intro: read the first paragraph in 8.3
Phase 1: read pp 439442 and review probability, do 8.3/14
Phase 2: finish reading 8.3, do all odd problems, find further reading
Phase 3: read further reading and do all even problems
Phase 4: do problems from the further reading

LU Factorization (Comp Sci Majors)
Intro: read the first paragraph in 9.3
Phase 1: read 9.2 and do exercises 9.2/1,3,5
Phase 2: do all odd exercises in 9.2
Phase 3: read 9.3 and do exercises 1, 3, 5
Phase 4: complete all odd exercises in 9.3

Linear Economic Models
Intro: read the first page in 8.5 and Example 1
Phase 1: read 461464, do exercises 1, 3 and 5
Phase 2: finish 8.5, get further reading from a library and do all
odd exercises
Phase 3: read further reading and do all problems
Phase 4: do problems from further reading

Linear Programming
Intro: read the first page in 7.1 and Example 1
Phase 1: read 7.1 pages 371379 and do exercises 1, 3, 11, 13
Phase 2: finish reading 7.1 and do all odd exercises
Phase 3: read 7.2 pages 389395 and do exercises 1,3
Phase 4: finish 7.2 and complete all odd exercises
(in future: 7.2/5, 7, 15

Graph Theory
Intro: read the first page in 8.1 and Example 1
Phase 1: read 8.1 pages 417422 and do exercises 1, 3, 5
Phase 2: finish 8.1 and do all odd problems
Phase 3: do problems 8.1/ 2,4,6,8, T.2, T.3
Phase 4: complete all even problems