## 403/503 – Dimension

February 1, 2010

I refer to the textbook for the basic notion and properties of vector spaces. A field is a triple ${{\mathbb F}=(F,+,\times)}$ satisfying the axioms listed in pages 2, 3 of the textbook as properties of ${{\mathbb C},}$ see also https://andrescaicedo.wordpress.com/2009/02/11/305-4-fields/ and surrounding lectures, in this blog. I am writing this note mostly to record the exercise, the question, and the statement of Steinitz lemma, so I am not recording the proofs we discussed in lecture.

The following example, that I want to leave as a (voluntary) exercise, is due to John Conway.

Exercise 1 Define Nim-addition ${\oplus}$ and Nim-multiplication ${\otimes}$ on ${{\mathbb N}}$ as follows:

1. ${n\oplus m}$ is the result of adding without carrying the binary expansions of ${n}$ and ${m}$. For example, ${9\oplus1=1001_2\oplus1=1000_2=8.}$
2. ${n\otimes m}$ is computed by applying the following rules:
• ${\otimes}$ is commutative.
• ${\otimes}$ distributes over ${\oplus.}$
• Letting ${F_n=2^{2^n},}$ we have $F_n\otimes F_n=\frac32F_n$ and ${F_n\otimes m=F_n\times m}$ for ${m

Show that ${({\mathbb N},\oplus,\otimes)}$ is a field.

## 403/503 – Syllabus

February 1, 2010

Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 1:40-2:30 pm.
Place: Mathematics/Geosciences building, Room 124.
Office Hours: MF 10:40-11:30 am.
Text: Axler, Sheldon. Linear algebra done right. Springer, 2nd edition (1997).

Contents: Math 403/503 is intended to be a second course in linear algebra, where an abstract approach emphasizing the role of linear transformations is preferred to a more computational approach based on properties of matrices. From Course Description in the Department’s site:

Concepts of linear algebra from a theoretical perspective. Topics include vector spaces and linear maps, dual vector spaces and quotient spaces, eigenvalues and eigenvectors, diagonalization, inner product spaces, adjoint transformations, orthogonal and unitary transformations, Jordan normal form..

Grading: Based on homework. Homework is due a week after it is stated, and no late homework is allowed. If it proves necessary, we will have a final exam at the end of the course.

Problem 2 is False. This is because we can have $x=-y\ne0,$ in which case $x^2=y^2$ but $x\ne y.$ For example, consider $x=1,$ $y=-1.$
For problem 3, start by writing the number $n=1\times 2\times \dots\times 9$ as a product of primes: $n=2^7\times 3^4\times 5\times 7.$ Plainly, any positive divisor of $n$ must have the form $2^a\times 3^b\times 5^c\times 7^d$ where $a=0,1,\dots,$ or $7;$ similarly, $b=0,1,\dots,$ or $4$; $c=0$ or $1,$ and $d=0$ or $1.$ There are 8 possibilities for $a,$ 5 for $b,$ 2 for $c,$ and 2 for $d.$ This gives us a total of $8\times 5\times 2\times 2=160$ possible positive divisors.