Math 108A - Intro to Linear Algebra - Spring 2008

Professor: Alex Dugas my homepage
Office: 6510 South Hall
Office Hours: M 11 - 12, T 2:30 - 4

Prerequisites: Math 5A, 8 (with a grade of C or better).

Texts: Sheldon Axler, Linear Algebra Done Right. Springer 1997.
Alternative Text (Recommended): Sergei Treil, Linear Algebra Done Wrong.

Lecture: MWF 12:00 - 12:50 pm. in Arts 1251.

The GSI for this course is Charles Martin.  His office hours are:

• MW 1 - 2 pm , 6432F
• Th 5 - 7 pm , Mathlab  1607 SH

Announcements:

• My Office Hours next week will be Monday 2:30 - 4:30.
  
• The Final Exam is next Tuesday 6/10 12-3pm.  It will be cumulative, but will focus slightly more on Chapter 5 of LADR.  Here is an old exam that you may practice.  The most important things for you to review are the Definitions and Major Theorems:
• Defintions: Subspace, Sum, Direct Sum, Span, Linear (In)Dependence, Basis, Dimension,  Linear Map, Null Space, Range, Invertible, Matrix of a Linear Map, Eigenvalue, Eigenvector, Eigenspace, Diagonalizable.
• Major Theorems: 2.10, 2.11, 2.12, 2.14, 2.15, 2.16, 2.17; 3.2, 3.4, 3.18; 5.6, 5.10, 5.13, 5.21(a,b).
• Of course, this does not mean these are the only topics that may appear on the test.  But these are definitely the most important.
  
• Solutions to Midterm 2.
  
• Here is Midterm 2.  It is due next Friday, May 23, by 4pm.
  
• If you want to improve your midterm 1 score, do this optional homework.  Please read it carefully and follow all instructions.  It is due by Friday 5/2.  To get your midterm back, contact the TA Charles Martin.
  
• Midterm 1 will cover chapters 1 and 2 of LADR. 
  
• The most important topics include
  
1. Vector spaces (know the definition and how to tell if something is a vector space or not.)
2. Subspaces (Be able to prove that a subset is or is not a subspace.)
3. Sums and Direct Sums of Subspaces (Know the definitions.  Be able to compute the sum of two or more subspaces, and determine whether the sum is direct.)
4. Span of a set of vectors.
5. Linear (In)Dependence of vectors.
6. Bases for vector spaces  (Know the definition and how to find a basis.  Be able to show that certain vectors form a basis.)
7. Dimension of vector spaces (Know how to compute the dimension of a vector space, and understand how this relates to bases, spans and linearly independent sets of vectors.)
8. As a practice Exam, do exercises 2.2, 2.3 and 2.6 on p. 10 of LADW and exercise 8 on p. 19 of LADR. Practice Exam Solutions
   

• Homeworks and readings are displayed in the table below.  "R" refers to the text "Linear algebra done RIGHT", and "W" refers to the text "Linear algebra done WRONG" (see the link above).  Please note that page numbers in W refer to the numbering of the text, and NOT of the pdf file.
• Each of you should come to my office hours at least once during the first three weeks of class to introduce yourself.  If the scheduled times do not fit your schedule, please email me or talk to me after class to arrange another time.
  

Course Timetable (subject to change)
Date	Topics	Reading    R = LADR W = LADW (-) = Optional	Homework
M 3/31	Intro / Review		Homework 1
W 4/2	Complex Numbers	R: p. 2-3
F 4/4	Vector Spaces	R: p. 4-10
M 4/7	Properties of Vector Spaces.  Subspaces.	R: p. 11 - 14	Homework 2
W 4/9	Sums of Subspaces.  Span.	R: p. 14
F 4/11	Direct Sums.	R: p. 15 - 18
M 4/14	Linear Independence.	R: p. 22 - 25
W 4/16	Dimension.	R: p. 25 - 27	Homework 3
F 4/18	Bases.	R: p. 28 - 32
M 4/21	Equivalent Characterizations of Bases.
W 4/23	Dimensions of sums of subspaces	R: p. 33-34
F  4/25	Midterm 1
M 4/28	Linear Maps.	R: p. 38-41	Homework 4: R: p. 59-60:  Ex. 2, 4, 5, 7, 9, 12
W 4/30	Null Space of a linear map.	R: p. 41-43
F 5/2	Range of a linear map.	R: p. 43-44
M 5/5	Rank-Nullity Theorem	R: p. 45
W 5/7	Systems of linear equations.	R: p. 46-47	Homework 5
F 5/9	Invertibility.  Isomorphisms.	R: p. 54-55
M 5/12	Matrices associated to linear maps.	R: p. 48-50
W 5/14	Matrices and linear maps (cont.)
F 5/16	Composition of linear maps and matrix multiplication.  Invertible matrices.	R: p. 51-53
M 5/19	Change of Bases.	R: p. 214-216
W 5/21	Eigenvalues, Eigenvectors, Eigenspaces	R: p. 76-79
F 5/23	Invariant Subspaces. Upper-Triangular Matrices	R: p. 82-84
M 5/26	Memorial Day Holiday: No Class		Homework 6   LADR: p. 94-5:    5, 7, 8, 9, 10, 11
W 5/28	Existence of eigenvalues over C.  Upper-triangular form of an operator.	R : p. 81, 84 - 85
F 5/30	Eigenvalues of upper-triangular matrices.	R: p. 85 - 87
M 6/2	Diagonal matrices.  Diagonalization.	R: p. 87 - 90
W 6/4	Invariant subspaces of R-vector spaces	R: p. 91 - 93
F 6/6	Spectral theory using Determinants (Summary.)
Tu 6/10	Final Exam - 12:00 - 3:00 pm

Course Content and Goals:  We will cover Chapters 1,2,3 and 5 in LADR (vector spaces, linear transformations, eigenvalues and eigenvectors) and additional topics from LADW (inverse matrices, determinants, change of basis) as time allows.  LADR emphasizes abstract properties of finite dimensional vector spaces and linear transformations, while LADW presents a more concrete matrix oriented approach.  I will try to combine these two perspectives, often asking you to read the texts side by side to compare.  In terms of proof writing, this course serves as an extension of Math 8.  On the homeworks and exams, I expect you to further practice and develop your logical reasoning and proof writing skills.

Homework:  Homework exercises will be assigned in lecture and listed on the course webpage.  Homework will be due in lecture each Wednesday.  You may work together on homework problems; however, you must write up your answers individually.  You must show all your work and clearly explain your reasoning in order to receive full credit.  Late homeworks will not be accepted.  However, your lowest homework score will be automatically dropped.

Exams: There will be two in-class midterm exam on Friday April 25, 12:00--12:50 pm and Wednesday May 21, 12:00--12:50 pm.  Please arrive promptly.  The final exam is scheduled for Tuesday June 10, 12:00--3:00 pm.  The second midterm and/or final exam may be take-home (to be decided later).  The problems on the exams will be similar to those seen in class or on the homeworks.   No make-up exams will be given, except in extraordinary circumstances.  If you have a serious conflict with any of these exams or miss one for any reason, it is your responsibility to notify me immediately so that other arrangements may be made

Grades:  Grades will be computed from your scores on homeworks and exams as follows:  Homework = 25%, Midterms = 20% each, Final = 35%.  No letter grades will be assigned until the end of the semester, and the exact grading scale will be curved relative to the difficulty of the exams.  However, a 90% or above will guarantee you at least an A, an 80% will be at least a B, and 70% will be at least a C.