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\centerline{Mathematics 352 -- Abstract Algebra II}
\centerline{Discussion 1 -- Vector Spaces}
\centerline{Friday, February 8, 2008}
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\noindent
{\it Background}
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Recall from Linear Algebra that a vector space 
over a field $F$ is a set $V$ of ``vectors''
with a sum operation and a multiplication operation 
by scalars in $F$.  Today we want to review 
some of the properties of vector spaces and 
see how they apply to some new examples related
to the topics we have been studying in this course.
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{\it Discussion Questions}
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A)  General properties.
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\item{1)}  What are the properties of the sum and scalar 
multiplication operations that define the vector space 
structure?  Try to express these in as economical form 
as possible (this might look different from the form you 
saw in the Linear Algebra course).  
\item{2)}  What is a {\it basis} for a vector space $V$?  
\item{3)}  It is a very important general theorem about
vector spaces that every vector space over a field $F$
has a basis, and that all bases have the same cardinality.  
One nice proof of this fact in the case
that the vector space $V$ has some finite spanning set 
is based on the following lemma:
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\item{} {\it Lemma}.  Let $V$ be a vector space
that has a finite spanning set $S$.  
If $T$ is any linearly independent subset of $V$, then
$|T| \le |S|$.   
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\item{}  Prove the lemma and use it to deduce that
if $B_1$ and $B_2$ are both finite sets that are bases
for a vector space $V$, then $|B_1| = |B_2|$.  Where 
exactly are you using that the scalars come from a field?
\item{4)}  What important invariant of vector spaces
is defined on the basis of what you showed in part 3?
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\noindent
The field $K$ that is part of a vector space structure
can be any field -- in particular, this means that 
examples such as $K = {\bf Z}_p$ or even 
$K = F[x]/\langle p(x)\rangle$ where 
$p(x)$ is an irreducible polynomial in $F[x]$ for a field
like $F = {\bf Z}_p$ are possibilities!
Moreover, the elements of a vector space do not necessarily
look like ``vectors'' in the usual sense of an ordered
$n$-tuple of elements of the field.  
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\noindent
B)  Some new examples of vector spaces.
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\item{1)}  Show that the set of all polynomials in two variables
$x,y$ with coefficients in $F = {\bf Z}_3$, of total degree $\le 3$
in $x$ and $y$ forms a vector space over the field $F$.  What 
is the dimension of this vector space?
\item{2)}  Let $K = {\bf Z}_2[x]/\langle x^3 + x + 1\rangle$
Show that $K$ forms a vector space over the field ${\bf Z}_2$.
What is the dimension of this vector space?  
\item{}  Note:  $p(x) = x^3 + x + 1$
is irreducible in ${\bf Z}_2[x]$. This implies, by a theorem from 
last semester, that $V$ is a field.  {\it A field $K$ can be 
a vector space over another field $F$(!)} We will be using
this observation extensively!  Another similar example,
\item{3)}  Let 
$$V = {\bf Q}[\sqrt{3}] = \{a + b\sqrt{3} : a,b \in {\bf Q}\}.$$
Show that $V$ is a vector space over $F = {\bf Q}$ and determine
its dimension.
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{\it Assignment}
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Writeups due in class on Wednesday, February 13.
\bye
