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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 | $n\equiv\mathrm{card}(B')$ |
| $v_1,\dots,v_n\in B'$ |
| $c_1,\dots c_n\in \mathcal F$ |
| $x\in V$ |
| 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
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