Vector space basis


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.


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.


Wikipedia: Vector space



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