
\documentclass{article}
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\begin{document}

%% Do not change the above and do not add your own definitions %%



\title{Vector spaces}
\author{Benne de Weger}
\date{}
\maketitle



\subsection{Fields}
A {\em field} $F$ is a set containing elements 0 and 1
in which the four operations addition, subtraction,
multiplication and division (not by 0) are defined
and satisfy the usual rules.
Common examples are the field $\mathbb{R}$ of real numbers,
the field $\mathbb{C}$ of complex numbers,
%[BdW] weggehaald, Q komt verder niet voor
%the field $\mathbb{Q}$ of rational numbers (fractions $p/q$
%with integral $p$ and integral nonzero $q$),
and the finite field $\mathbb{F}_p$ of integers modulo $p$
for some prime number $p$.

In analysis the underlying field will often be $\mathbb{R}$ or $\mathbb{C}$.
In coding theory one uses the field $\mathbb{F}_2 = \{0,1\}$ of two elements.
%[BdW] weggehaald, weinig relevant voor het vervolg
%Instead of $\mathbb{F}_q$ one also uses the notation $GF(q)$ (`Galois field')
%for a finite field with $q$ elements. (Up to isomorphism there is a unique
%such field for each prime power $q$.)

%[BdW] weggehaald
%Exercises
%
%- compute $5+6$ in the finite field $\mathbb{F}_7$
%
%- is the set $\mathbb{Z}$ of integers (with the usual operations) a field?

\subsection{Vector spaces}
A {\em vector space} $V$ (over a field $F$) is a set, the elements
of which are called {\em vectors}, such that one can add (and subtract)
vectors and multiply a vector by a constant from $F$.
%[BdW] edit
%(And these constants are called {\em scalars}.)
These constants are called {\em scalars}.

\medskip
Formally, the axioms are:
(i) $V$ is an abelian group ($v+w = w+v$ and $u+(v+w) = (u+v)+w$
and $u+0=0+u=u$ and $u-u=0$ for all $u,v,w$ in $V$), and
(ii) $V$ admits a scalar multiplication by elements of $F$, such that
$-u = (-1)u$ and $a(u+v) = au+av$ and $a(bu) = (ab)u$ for all
$a,b$ in $F$ and $u,v$ in $V$.

\medskip
For example, $\mathbb{R}^n$ becomes a vector space (over $\mathbb{R}$)
for the operations
$(x_1,...,x_n) + (y_1,...,y_n) = (x_1+y_1,...,x_n+y_n)$ and
$a (x_1,...,x_n) = (ax_1,...,ax_n)$ for all $a$ and $x_i$ in $\mathbb{R}$.
And similarly for $\mathbb{C}^n$, where now the $x_i$ and the constants
are from $\mathbb{C}$.

\medskip
A more complicated example would be the space $C[0,1]$ of all real-valued
continuous functions defined on $[0,1]$ with the natural operations
$(f+g)(x) = f(x)+g(x)$ and $(af)(x) = af(x)$.

\subsection{Subspaces}
A subset $W$ of a vector space $V$ is called a {\em subspace}
when it becomes a vector space with the operations inherited from $V$,
that is, when sums and scalar multiples of vectors in $W$ again lie in $W$.

The intersection of subspaces is again subspace.

\medskip
If $A$ is a set of vectors in $V$, then the {\em span}
$\langle A \rangle$ of $A$ is the smallest subspace of $V$ that
contains $A$. It can be constructed by taking all
{\em linear combinations} (that is, sums of scalar multiples)
of vectors in $A$.

\medskip
A set of vectors is called (linearly) {\em independent} when none of them
lies in the span of the others, and (linearly) {\em dependent} otherwise.
%[BdW] Dus {0} is onafhankelijk?

\subsection{Basis and dimension}

A collection of vectors $B$ in a vector space $V$ is called a {\em basis}
when each vector in $V$ can be written in a unique way as linear
combination of elements of $B$. The bases are precisely the maximal
independent sets of vectors. In particular, every independent set
can be enlarged to a basis. 
%[BdW] zin toegevoegd
The bases are also precisely the minimal spanning sets of vectors,
and in particular, every spanning set can be reduced to a basis. 
Every vector space has a basis.

\medskip
Any two bases of $V$ have the same size (cardinality), and this size
is called the {\em dimension} of $V$,
%[BdW] notatie toegevoegd, en zin daarna verkort
notation: $ \dim V $. For example, $ \dim \mathbb{R}^n = n $, and
$ \dim C[0,1] = \infty $.
%For example, $\mathbb{R}^n$ has dimension $n$, and $C[0,1]$ is
%infinite dimensional.

\medskip
For example, the {\em standard basis} of $\mathbb{R}^n$
consists of the $n$ vectors $e_1 = (1,0,...,0)$, ...,
$e_i = (0,...,0,1,0,...,0)$, ..., $e_n = (0,...,0,1)$,
where $e_i$ has a 1 in the $i$-th position.

\medskip
Given a basis $B = \{e_i \mid i \in I \}$, the {\em coordinates}
of a vector $v$ (when written on the basis $B$)
are the numbers $a_i$ such that $v = \sum_i a_i e_i$.

\medskip
For example, in $\mathbb{R}^2$ one has the standard basis $\{e_1,e_2\}$
with $e_1 = (1,0)$ and $e_2 = (0,1)$. Written on this basis the vector
$(2,1)$ has coordinates $2,1$.
A different basis is $\{(-1,1),(1,1)\}$. Written on that basis the same
vector has coordinates $-\frac{1}{2},\frac{3}{2}$, since
$(2,1) = -\frac{1}{2}(-1,1) + \frac{3}{2}(1,1)$.

\subsection{Quotient spaces}
If $V$ is a vector space, and $W$ a subspace, then the {\em cosets}
of $W$ are the translates $v + W = \{ v+w \mid w \in W \}$.
The collection of cosets becomes a vector space itself
under the natural operations $(u+W)+(v+W) = (u+v)+W$ and
$a(u+W) = au+W$. This vector space is called the {\em quotient space}
$V/W$. The {\em codimension} of $W$ in $V$ is $\dim V/W$.
If $V$ is finite-dimensional, then $\dim V/W = \dim V - \dim W$.

\subsection{Direct sums}
%[BDW] voorstel voor andere tekst
%reden: ik vind het prettiger om formele optellingen hier
%te vermijden (die heb je al bij het Cartesisch product),
%en eerst de gewone som in te voeren. 
If $W_1$ and $W_2$ are subspaces of a vector space $V$, then the {\em sum} 
$W_1 + W_2$ is the vector space of which the vectors are
sums $w_1+w_2$ with $w_1 \in W_1$ and $w_2 \in W_2$.
One has $\dim (W_1 + W_2) \leq \dim W_1 + \dim W_2$.
If in addition $ W_1 \cap W_2 = \{ 0 \} $ then we speak about
the sum as the {\em direct sum} $W_1 \oplus W_2$.
Then $\dim (W_1 \oplus W_2) = \dim W_1 + \dim W_2$.

%If $V$ and $W$ are vector spaces, then the {\em direct sum}
%$V \oplus W$ is the vector space of which the vectors are
%formal sums $v+w$ with $v \in V$ and $w \in W$ with the natural operations.
%One has $\dim (V \oplus W) = \dim V + \dim W$.

\medskip
If $V$ and $W$ are vector spaces, then the {\em Cartesian product}
$V \times W$ is the vector space of which the vectors are
the pairs $(v,w)$ with $v \in V$ and $w \in W$ with the natural operations.
One has $\dim (V \times W) = \dim V + \dim W$.
Of course `direct sum' and `Cartesian product' are different names
for the same thing.

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