University of California, Berkeley

MWF 1pm-2pm, 155 Dwinelle (Lecture)

MWF 4pm-5pm, 71 Evans Hall (Discussion Section 210)

MWF 5pm-6pm, 71 Evans Hall (Discussion Section 211)

** Instructor: ** Prof. Dan-Virgil Voiculescu

** GSI: **

- Daniel Appel (1058 Evans)
- David Berlekamp (739 Evans)
- Harrison Chen (1058 Evans)
- Chang-Yeon Cho (1062 Evans)
- Gregory Igusa (775 Evans)
- George Melvin (853 Evans)

** Email: **

** Office: ** 853 Evans Hall

** Office Hours: ** Monday 6-8pm, Friday 3-4pm - 853 Evans Hall

** Content: ** From the online schedule: *Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations. *

** Textbook: ** Linear Algebra & Differential Equations (UC Berkeley Custom Edition), * Lay; Nagle, Snaff, Snider* (Here is the first chapter of the book for those of you who have yet to obtain the textbook)

** Syllabus: ** Additional course information, including the grading policy for the course.

** Exams: ** There will be TWO midterms and ONE final exam for this course. The grading policy for exams may be found in the additional course information above. * BE AWARE OF THE POLICY CONCERNING MISSED MIDTERMS! *

- Midterm 1: Monday, 24th September 2012 (Solutions)
- Midterm 2: Monday, 29th October 2012 (Solutions)
- Final: Wednesday, 12th December 2012

** piazza.com ** I have started a class forum at piazza.com, if you would like to added to this then please send me an email. Use this forum to ask any questions you have concerning the material covered during class and on any problems you have with homework; also, feel free to * answer* any questions you feel comfortable discussing with your fellow students. Try to be civil with each other!

** Please Note: ** the invitation to piazza.com is only for those students who are enrolled in my (=George's) sections.

** Notes: ** Here is an example on determining the solution set of an arbitrary matrix equation. This method will be useful later in the course.

Here is a short review sheet for Midterm 1.

Here is a short review of polynomial vector spaces.

Here is a short review of least squares problems.

Here is a review of the diagonalisation process.

Here is a review of linear maps and matrices.

Here is a review of orthogonal diagonalisation and the main ideas from Section 7.1 of the textbook.

Here is a short review on finding complex eigenvectors and their use in solving systems of differential equations.

Here is a short review on the general theory of linear ordinary differential equations.

Here is a review of Fourier series and their convergence properties.

Here is a review of solving the heat equation with 'formal solutions'. (*Updated 12/9: the formula for the 'formal solution' was incorrect and has been updated*)

** Practice Exams: ** Here are some old midterms for previous Math 54 courses (*Note: injective = one-to-one; surjective = onto *):

Previous Tests courtesy of some fraternity.

- Fall 06 (Voiculescu)
- Fall 09 (Voiculescu)
- Summer 2010 (Melvin)
- Summer 2011 (Melvin)

** Extras: ** Here is some useful information on problem solving techniques given by the Hungarian mathematician George Polya.

** History: ** Here is an interesting article about the German mathematician Hermann Grassmann and his involvement in the development of linear algebra during the 19th Century.

** Homework is due on Wednesdays during the semester at the beginning of discussion section - 4.10pm or 5.10pm. Homework assignments will be posted below. Homework will be graded on completeness and correctness. **

Late homework will not be accepted.

If you are unable to submit your homework at the required time then you can leave it in my mailbox (situated in the 9th floor mail room of Evans Hall, opposite the North Elevators) before it is due. **Please email me if you intend to leave your homework in my mailbox**.

Collaboration on homework is welcome and encouraged although if you are working with another student please state that you have done so (eg. if you work with A. Nother on a particular question just write "This question was completed with A. Nother."). However, all homework assignments must be written up individually. Failure to declare collaboration with another student will result in a grade penalty (and it is remarkably simple to tell when students have copied each other). Also, if you have used a textbook or online notes to help you understand/solve a problem please cite a reference (eg. if you used pages 52-60 of Prof. X's online lecture notes just write "This question used p.52-60 of Prof. X's online lecture notes, available at www.math.com/~profx/linalg)

* References: LA: Linear Algebra part of textbook; DE: Differential Equations part of textbook.*

** Homework Assignments: **

** Reading Assignments: **

There will be a quiz during discussion section on each Monday following a homework assignment. Each quiz will be 20 minutes. Solutions to quizzes will be posted below as soon as possible.

There will be no make-up quizzes.

** The first quiz will take place during discussion section on *Wednesday, 5th September* as there is no class on Monday 9/3. **

** Solutions **

Here are the worksheets that are handed out during discussion section. You should use these worksheets to get extra practice at computations. They will also highlight various consequences of Theorems you will see in this course. If you have any questions on the worksheets then please get in contact with me; better still, ask a question at piazza.com!

- Worksheet 8/24
- Worksheet 8/27
- Worksheet 8/29
- Worksheet 8/31
- Worksheet 9/7
- Worksheet 9/10
- Worksheet 9/12
- Worksheet 9/17
- Worksheet 10/1
- Worksheet 10/3
- Worksheet 10/8 (
*In Q3iii) there is a mistake - the given p is not in H! Take p=2X-2X^{2} and change the equations of H to p(0)=p(1) and p(-1)=p(2)*) - Worksheet 10/10
- Worksheet 10/12
- Worksheet 10/17 (
*Note: in Q4 it should read '... c_{1}=2b_{2}...'*) - Worksheet 10/22
- Worksheet 10/24
- Worksheet 11/2
- Worksheet 11/5
- Worksheet 11/19
- Worksheet 11/21
- Worksheet 11/26
- Worksheet 11/30

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