Vector Spaces
Vector spaces allow us to collect together vectors with similar characteristics.
Basics
We have $U$ - a set of $n$-vectors using $H$:
- e.g. $U$ is the set of all 3-vectors from $\Bbb R$.
The set $U$ defines a vector space if it has two closure properties:
- Adding any two vectors in $U$ produces a vector in $U$.
- Multiplying any vector in $U$ by a constant in $H$ produces a vector in $U$.
Formal Definition
Given $U$ - a set of $n$-vectors using $H$. The set $U$ defines a vector space if: \(\begin{align} \forall v\in U, \forall w\in U &:v+w\in U &\text{(C1)}\\ \forall v\in U, \forall \alpha\in H &: \alpha v\in U &\text{(C2)} \end{align}\)
The underlying set $H$ is very important.
Examples
-
$\text{EVEN}=\{\langle x,y \rangle:x+y=2p,p\in\Bbb Z\}$
If $H=\Bbb Z_k$ then $\text{EVEN}$ is a vector space.
If $H=\Bbb R$ then $\text{EVEN}$ is not a vector space.
- Consider C2 using $\langle1,1\rangle,\alpha=\frac{1}{2}$.
-
$\text{ODD}=\{\langle x,y\rangle:x+y=2p-1,p\in\Bbb Z\}$
For this $\text{ODD}$ is not a vector space e.g. $H=\Bbb Z _{16}$:
- $\langle1,4\rangle\in\text{ODD},\langle4,1\rangle\in\text{ODD}$
- $\langle5,5\rangle\notin\text{ODD}$
Attributes
Suppose $U$ is a vector space of $n$-vectors from $\Bbb R$. Important notions are a basis set and its dimension.
The idea of basis allows us to reduce $U$, even if it is infinite, to at most $n$ distinct vectors. We do this by expressing members of $U$ as linear combinations of other vectors in $U$.
Linear Combinations
Take a set $B$ of $m$ vectors from $U$:
\[B=\{t_1,t_2,\ldots,t_m\}\]Take a collection $\vec a$ of $m$ values at least one of which is not 0 and all non-zero values are from $H$.
\[\vec a=\langle\alpha_1,\alpha_2,\ldots,\alpha_m\rangle\]$\vec w$ is a linear combination of $B$ using $\vec a$ is $\vec w$ is:
\[\sum^m_{k=1}\alpha_kt_k\]Linear Dependence & Independence
Now suppose that $U$ is a vector space and $B$ is a subset of $k$ $n$-vectors from $U$ for which any $w$ in $U$ can be obtained as a linear combination of $B$?
That is:
\[\forall w\in U\exists\langle\alpha_1,\alpha_2,\ldots,\alpha_k\rangle:w=\sum^k_{i=1}\alpha_ib_i\]Then only $B$ is needed to capture $U$.
Basis and Dimension
A subset $B$ of $U$ for which any vector in the vector space $U$ can be formed as a linear combination of $B$ is called a basis of $U$.
The number of vectors in the smallest basis of $U$ is called the dimension of $U$ denoted $\text{dim}(U)$.
A vector space $U$ of $n$-vectors have a dimension of at most $n$.
Example
\[\text{EVEN}=\{\langle x,y \rangle:x+y=2p,p\in\Bbb Z\}\]We saw that this was a vector space for $H=\Bbb Z_4$.
It contains:
\[\left\{ \begin{align} \langle0,0\rangle, & \langle0,2\rangle, & \langle1,1\rangle, & \langle1,3\rangle,\\ \langle2,0\rangle, & \langle2,2\rangle, & \langle3,1\rangle, & \langle3,3\rangle \end{align} \right\}\]$\text{dim(EVEN)}=2;B=\{\langle0,2\rangle,\langle3,1\rangle\}$
e.g. $\langle1,1\rangle=1 * \langle0,2\rangle+3 * \langle3,1\rangle$