*Warning:* This course will be taught quite theoretically, and
at a fast pace! As a result, the homework will
be heavy too. If you
are interested mainly in applications to other fields,
or prefer to deal only with concrete objects, you should
take Math 103 instead.

**Lectures:**
Tuesday and Thursday 11 am - 12:15 pm.
Location as of Thursday January 12: 380-Y.

**Professor:** Ravi Vakil,
Room 383-M (third floor of math building).

**Course assistant:** Robin Koytcheff, robink-at-math.stanford.edu,
Room 381-N (first floor of math building).

**Course graders:** Theo Johnson-Freyd, theojf-at-stanford.edu,
and Charles Hallford, hallford-at-stanford.edu. (If you have
questions about grading, please contact Theo.)

**Office hours:**
Monday 2:05-4:05 pm (Robin Koytcheff, 381-N),
Tuesday 1-2 pm (Robin Koytcheff, 381-N),
Tuesday 2-3:30 pm (Ravi Vakil, 383-N) **(except March 14)**,
Wednesday 2:05-3 pm (Robin Koytcheff, 381-N).
Office hours on the week of the final exam are announced below.
There will be three hours on each of Sunday March 19, Monday March 20,
and Tuesday March 21, in either the afternoon or the evening.

**Text:** The textbook for the course will be the notes *A
(terse) introduction to linear algebra*, by Prof. Katznelson (who
taught this class in the fall).

You are certainly encouraged to look at other references, to find
one that suits your point of view. The best way to do this is to
browse in the library. Here are two books
that some people like:
Hoffman and Kunze, *Linear Algebra*; and
Nering, *Linear Algebra and Matrix Theory*.
Both should be on 3-day reserve at the Math and CS library.
Another book recommended by Robin is Axler's
*Linear Algebra Done Right*.
*Warning:* The bookstore has "Hoffman and Kunze" advertised
as the text for this class. This is not the case!

**E-mail list:**
I will maintain an e-mail list to sporadically send you important messages.
If you are not on the e-mail list and wish to be, please let me know.
(And similarly, let me know if you wished to be removed.) This is *not* the same list as those
officially enrolled on Axess.

**Grading:**

**What we'll cover:**
We will hopefully cover most of the following:
Chapters 1, 2 (emphasizing 2.6 less), 3, 4.
Chapter 5 up to 5.3. Hopefully 6.1, and perhaps more of 6.
As much of the appendix as we need. We will largely move linearly
through the text, so you should read one section ahead of where we are.

**What we've done so far:**
(This paragraph will be only sporadically updated.)

Week 1: We discussed maps, groups, and fields. We went through
1.1, except for tensor products.

Week 2: We discussed until the end of chapter I.

Week 3 and
Week 4: We discussed up until change of basis (II.4.5).
The midterm took place in week week 4.

Week 5: We skipped 2.6, and finished most of chapter 3.

Week 6 and 7: We discussed determinants (chapter 4), with topics in a slightly
different order than in the book.

1. (A first proof problem.) Show (using the axioms of a vector space) that 0 times any vector

2. Recall our definition of quotient space V/W, where W is a subspace of vector space W. We defined V/W as a set, and claimed that it is in fact a vector space. We showed how to multiply an element

Do problems I.1.1, I.1.2, I.1.3, I.1.4, I.1.5, I.2.1, I.2.2, I.2.3, I.2.6, I.2.7 in the text. (All references are to the January 1 version.) You can do I.1.6 in place of any three of the above questions; and you do I.1.7 in place of any three as well. (In I.1.5, I.1.6, and I.2.2, that backslash symbol means "minus" for sets. In I.1.6, you are required to verify the first sentence. In I.1.7 (a), you are required to verify the statement.) The I.2 problems will require reading ahead.

1. Show that any two vector spaces over a field F of dimension 10 are isomorphic.

I.2.9, I.2.12, I.2.15, I.3.1 (explain why you have found all such matrices), I.3.2, I.3.3 (define "degrees of freedom" in any sensible way), I.3.5, I.3.7, I.3.9. You can do I.2.17 in place of any three above the above questions; and you can do I.3.10 in place of any three as well.

II.1.1, II.1.2, II.1.3, II.3.2, II.3.3, II.3.5, II.3.12, II.4.2. If you scored less than 4/6 on the last problem on the midterm, you can solve that problem in place of any two of the above.

Do problem 1, as well as seven of the remaining nine problems. (Or if you'd like, do all the problems, and the lowest two scores, not including problem 1, will be dropped.)

1. Suppose T: V --> W is a linear map, and T*: W* --> V* is the adjoint. Explain why (T

2. Let A be the 2x2 matrix with entries 2, 1 (in the top row), 1, 2 (in the bottom row). (a) Find 2 non-zero vectors

3. (In this problem, the row vectors should be read as column vectors. I hope this isn't too confusing!) Notice that

4. Consider the linear map F^{100} --> F^{101} given by

II.3.13, II.5.1, II.5.2, II.5.9, III.1.1, III.1.2.

III.1.7, III.2.1, III.2.3, IV.1.1, IV.1.2, IV.2.2, IV.5.3, IV.5.9.

IV.4.2 (This will involve figuring out what T-invariant means. Possible hint for the problem: choose a convenient basis. What does the matrix corresponding to T look like with respect to this basis?). IV.5.1. IV.5.2. IV.5.5. IV.5.8 (the second sentence should read: "In other words, if the characteristic polynomial of (a_{i,j}) is equal to (-1)^n product_{j=1}^n (lambda - lambda_j) then sum lambda_j = sum a_{i,i}."). IV.5.10 (this is an example of a "sparse" matrix, in which most entries are zero, and finding the determinant requires less compouting power). IV.5.11. IV.5.12 (feel free to add the additional assumption that the alpha_j's can be made, although strictly speaking it isn't necessary).

1. Prove part (c) of the Spectral Mapping Theorem (p. 66). (Don't just copy out the proof in the book --- it is a bit telegraphic! Piece together a proof that you understand.)

2. The next three problems will let you see why taking powers of diagonal and more generally upper-triangular matrices is not so hard, and why that makes taking powers of general matrices not so hard too. Consider the 2x2 diagonal matrix with diagonal entries 2 and 3. Then computing its 100th power is easy: it is the diagonal matrix with entries 2^{100} and 3^{100}.

(a) Suppose C is an invertible

(b) Let B be the 2x2 matrix

0 1

1 1.

If the field F is the field of rational numbers Q, show that B cannot be diagonalized. Diagonalize B over the real numbers R. (This will be a bit messy.)

3. (a) Using 2(a) and the diagonalization in 2(b), give an expression for B^n.

(b) Recall that the Fibonacci numbers are defined by F_0 = 0, F_1 = 1, and for n at least 2, F_n = F_{n-1} + F_{n-2}. Give an expression for B^n in terms of Fibonacci numbers. (Hint: first compute B, B^2, B^3, B^4, and look for the pattern.) Hence using 3(a), give a formula for the nth Fibonacci number! (Hint: the square root of 5 will appear in your formula.)

4. What is the top right entry in the 100th power of the 3x3 matrix A, given by

2 1 0

0 2 1

0 0 2?

(Possible hint: first try the same problem with a new matrix A', with 1's in the diagonal instead of 2's. Take the first few powers of A', and look for a pattern. Then take the first powers of A, and look for a pattern.) We will soon see that if you know the answer to this, you will be able to take any power of any matrix.

Do problems V.I.2, V.1.4, V.2.1, V.2.2.

V.2.4, V.2.5, V.2.6, V.2.7, V.2.10, V.2.11, V.3.1, V.3.6

Please come discuss the problems during office hours!

Here is the practice midterm. Here is the actual midterm. Some solutions are here (excepting B4, whose solution appears on problem set 3). Many other solutions are possible! I'm happy to discuss solutions to any of the problems.

**Bonus office hours:**
Sunday March 19 for 3 hours (Robin Koytcheff; the precise time will be announced here). Monday March 20, 7-10 pm (Ravi Vakil).
Tuesday March 21, 2-5 pm (Ravi Vakil). **GOOD LUCK!**

Ravi Vakil

Department of Mathematics Rm. 383-M

Stanford University

Stanford, CA

Phone: 650-723-7850 (but e-mail is better)

Fax: 650-725-4066

vakil/at/math.stanford.edu