Vector space basis

Set

context

$V$ …

$\ \mathcal F$ -vector space

definiendum

$B\in \mathrm{basis}(V)$

context

$B\subset V$

$B'\subseteq B$

$B'$ …finite

range

$n\equiv\left|B'\right|$

$v_1,\dots,v_n\in B'$

$c_1,\dots c_n\in \mathcal F$

$x\in V$

postulate

$\sum_{k=1}^n c_k\cdot v_k=0\ \Rightarrow\ \forall j.\ c_j=0$

All finite subsets of the base are linearly independed. It's maybe more clear when written in the contrapositive: “ $\exists j.\ c_j\ne 0\ \Rightarrow\ \sum_{k=1}^n c_k\cdot v_k\ne 0$ .”

postulate

$\exists c_1,\dots,c_n.\ (x=\sum_{k=1}^n c_k\cdot v_k)$

For each basis $B$ , every vector $x\in V$ has representation as linear combination.

Discussion

We call the vector space finite if it has a finite basis.

The difficulty in defining the basis of a general vector space above, and the reason why one must consider finite subsets $B'$ of the base $B$ , is that an infinite sum would require more structure than just what a general vector space provides (e.g. a metric w.r.t. which the series converges).

The zero vector space has an empty base. Its vector space dimension is zero.

Reference

Wikipedia: Vector space

Parents

Context

Vector space, Set cardinality, Finite sum over a monoid