Week 3: September 25-29, 2017.

A reminder about time and location of Peer Study Team (PST) sessions for this course: Thursday 4-6, UC257. The sessions are being run by Phil Blakey and Mira Kuroyedov.

The `course change date' for changing from MAT240F to MAT223F without academic penalty is October 3. See here for more information.

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On a related note: As announced in class today, we decided that the front row seat, right next to the center aisle, will from now on be reserved for one of your classmates who needs this arrangement.

Required reading: Re-read Chapters 1.1 and 1.2.

In class, we're bit behind and only now finished discussion of complex numbers (cf last week): Basic properties, how to take the inverse of a complex number, geometry of multiplication of complex numbers, polar coordinates, triangle inequality $|z+w|\le |z|+|w|$, fundamental theorem of algebra (without proof), factorizing polynomials. Definition of vector spaces, first examples.

A subfield of a field $F$ is a subset $F'\subset F$ containing $0$ and $1$, with the property that sums, products, additive inverses, and multiplicative inverses of elements in $F'$ are again in $F'$. In this case, $F'$ is itself a field (the field axioms for elements of $F'$ are imediate from those for $F$).

Two vector space $V,W$ over a given field $F$ are isomorphic , written as $$V\cong W $$ if there exists an identification of $V$ and $W$ as sets which also identifies the additions, scalar multiplications, and additive neutral elements. More precisely, this means that there exists an invertible function (bijection) $\phi \colon V\to W$ such that $\phi(u+v)=\phi(u)+\phi(v)$ and $\phi(tu)=t\, \phi(u),\ \phi(0_V)=0_W$. We write $\cong$ rather than $=$ because the identification, i.e. the map $\phi$ with these properties, is not unique. For example, if $S$ is a set with exactly $n$ elements, then the vector space $\mathcal{F}(S,F)$ of all functions $f\colon S\to F$ is isomorphic to $F^n$. Indeed, once we enumerate the elements of $S$ as $x_1,\ldots,x_n$, we obtain an identification by taking $f\in \mathcal{F}(S,F)$ to the $n$-tuple of function values $(f(x_1),\ldots,f(x_n))$. However, this identification depends on our choice of enumerating the elements of $S$; different choice give different identifications.

For a field $F$, and any given $n\in \mathbb{N}$ we introduced a vector space $\mathrm{Pol}_n(F)$ of all functions $p\colon F\to F,\ x\mapsto p(x)$ such that $p(x)=a_n x^n+a_{n-1} x^{n-1}+\ldots+a_1 x+a_0$. We also introduced a space $\mathrm{Pol}(F)$ of functions that can be written in this form for some $n$. If $F$ is a finite field, this space is different from the space $P(F)$ considered in the textbook; the latter consists of `formal' expressions $a_n x^n+a_{n-1} x^{n-1}+\ldots+a_1 x+a_0$ where $x,x^2,\ldots$ are just `formal variables'. For example, $P_n(F)=F^{n+1}$ since such a formal expression is uniquely given by the set of coefficients, $a_0,\ldots,a_n$. By contrast, in the definition of $\mathrm{Pol}(F)$ the $x$ it is an element of $F$.

Homework #2 is due 11pm on Friday, September 29

Additional homework (not to be handed in):

Do examples with long division of polynomials, finding roots.
Show that $\mathbb{Z}_2$ is a subfield of the field with $4$ elements.
The characteristic of a finite field $F$ is the smallest number $p$ such that $1+1+\ldots+1=0$ (with $p$ summands). Show that the characteristic is a prime number, and $\mathbb{Z}_p$ is a subfield of $F$.

The empty set is not a vector space. Why?