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\begin{document}

\title{Introduction to algebra --- handout 12}
\maketitle

\section{Vector spaces}

\begin{df}
Let $\langle F, +, \cdot\rangle$ be a field.  Let $\langle V, \oplus,
\odot\rangle$ is called a vector space over $F$ if $\oplus V\times
V\to V$ and $\odot F\times V\to V$ satisfying the following properties
\begin{enumerate}
\item $(v\oplus u) \oplus w=v\oplus (u\oplus w)$,
\item $v \oplus u=u\oplus v$,
\item $\exists o \enskip o\oplus v=v$,
\item $\exists -\!v\enskip v+(-v)=o$,

\item $\mu\odot(\nu \odot v)=(\mu\cdot \nu) \odot v$,
\item $\mu\odot(v\oplus u)=(\mu\odot v) \oplus (\mu\odot u)$,
\item $(\mu+\nu)\odot v=(\mu\odot v)\oplus (\nu\odot v)$, 
\item $1\odot v=v$.
\end{enumerate}
\end{df}

\begin{ex}${}$
\begin{enumerate}
\item $\R^n$; $\Q^n$; $\C^n$; $\Z_2^n$;
\item $\R^{n\times m}$;
\item polynomials; polynomials of degree at most $d$; 
\item functions; continuous functions; differentiable functions;
\item sequences; convergent sequences;
\item $\C$ over $\R$; $\R$ over $\Q$.
\end{enumerate}
\end{ex}

\begin{df}
Let $V$ be a vector space over field $F$.  Vectors $v_1$, \ldots, $v_n$ are called
{\bf linearly independent} if $\lambda_1\odot v_1\oplus\ldots\oplus
\lambda_n\odot v_n= o$ then $\lambda_1=\ldots=\lambda_n=0$. We say that vectors $v_1$,
\ldots, $v_n$ {\bf span} $W$ if for all $w\in W$ there are
$\lambda_1$,\ldots, $\lambda_n\in F$ such that $w=\lambda_1\odot
v_1\oplus\ldots\oplus \lambda_n \odot v_n$.  A {\bf basis} is a set of
linearly independent vectors spanning $V$.
\end{df}

\begin{thm}
If $b_1,\ldots, b_n$ and $c_1\ldots c_m$ are bases of the same vector
space, then $n=m$.
\end{thm}

\begin{df}
The {\bf dimension} of $V$, in symbols $dim\, V$, is the number of vectors of
a base.
\end{df}

\begin{thm}
Let $V$ be a vector space over field $F$. If $dim\, V=n$, then $V$ is
isomorphic to $F^n$.
\end{thm}

\begin{df}
Let $V$ be a vector space. $W$ is called a {\bf subspace} of $V$ if
$W$ is a vector space with the same operations.
\end{df}

\begin{thm}
Let $V$ be a vector space over field $F$. $W$ is a subspace iff the following holds:
\begin{enumerate}
\item $o\in W$;
\item if $v,u\in W$, then $v\oplus u\in W$; 
\item if $v\in W$ and $\mu\in F$, then $\mu\odot v\in W$. 
\end{enumerate}
\end{thm}

\begin{e}
Which of the following sets are subspaces of $\R^3$? (Why?)
\begin{itemize}
\item $W_1=\{(x,y,z)\in \R^3: x^2+y^2+z^2<1\}$,
\item $W_3=\{(x,y,z)\in \R^3: 3x+2=y+2z\}$,
\item $W_2=\{(x,y,z)\in \R^3: 2x+3y=z\}$,
\item $W_4=\{(x,y,z)\in \R^3: 2x-1=3y+2=z\}$,
\item $W_5=\{(x,y,z)\in \R^3: x,y,z\in\Q\}$.
\end{itemize}
\end{e}

\end{document}