===== Vector space basis =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $V$...$\ \mathcal F$-vector space |

| @#FFBB00: definiendum | @#FFBB00: $B\in \mathrm{basis}(V)$ |

| @#55CCEE: context     | @#55CCEE: $B\subset V$ |

| $B'\subseteq B$ | $B'$...finite | @#DDDDDD: range       | @#DDDDDD: $n\equiv\left|B'\right|$ |

| $v_1,\dots,v_n\in B'$ |
| $c_1,\dots c_n\in \mathcal F$ |
| $x\in V$ |

| @#55EE55: postulate   | @#55EE55: $\sum_{k=1}^n c_k\cdot v_k=0\ \Rightarrow\ \forall j.\ c_j=0$ |

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

| @#55EE55: postulate   | @#55EE55: $\exists c_1,\dots,c_n.\ (x=\sum_{k=1}^n c_k\cdot v_k)$ |

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 [[http://en.wikipedia.org/wiki/ero_vector_space#Vector_space|zero vector space]] has an empty base. Its [[vector space dimension]] is zero.
==== Reference ====
Wikipedia: [[http://en.wikipedia.org/wiki/Vector_space|Vector space]]
==== Parents ====
=== Context ===
[[Vector space]], [[Set cardinality]], [[Finite sum over a monoid]]