Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | Next revision Both sides next revision | ||
vector_space_basis [2013/09/07 19:31] nikolaj |
vector_space_basis [2014/03/21 11:11] 127.0.0.1 external edit |
||
---|---|---|---|
Line 1: | Line 1: | ||
===== Vector space basis ===== | ===== Vector space basis ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#88DDEE: $V$...$\ \mathcal F$-vector space | | + | | @#55CCEE: context | @#55CCEE: $V$...$\ \mathcal F$-vector space | |
- | | @#FFBB00: $B\in \mathrm{basis}(V)$ | | + | | @#FFBB00: definiendum | @#FFBB00: $B\in \mathrm{basis}(V)$ | |
- | | @#88DDEE: $B\subset V$ | | + | | @#55CCEE: context | @#55CCEE: $B\subset V$ | |
- | | $B'\subseteq B$ | $B'$...finite | @#DDDDDD: $n\equiv\mathrm{card}(B')$ | | + | | $B'\subseteq B$ | $B'$...finite | @#DDDDDD: range | @#DDDDDD: $n\equiv\mathrm{card}(B')$ | |
| $v_1,\dots,v_n\in B'$ | | | $v_1,\dots,v_n\in B'$ | | ||
Line 13: | Line 13: | ||
| $x\in V$ | | | $x\in V$ | | ||
- | | @#55EE55: $\sum_{k=1}^n c_k\cdot v_k=0\ \Rightarrow\ \forall j.\ c_j=0$ | | + | | @#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: "$\exists j.\ c_j\ne 0\ \Rightarrow\ \sum_{k=1}^n c_k\cdot v_k\ne 0$." | 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: $\exists c_1,\dots,c_n.\ (x=\sum_{k=1}^n c_k\cdot v_k)$ | | + | | @#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. | For each basis $B$, every vector $x\in V$ has representation as linear combination. | ||
Line 30: | Line 30: | ||
Wikipedia: [[http://en.wikipedia.org/wiki/Vector_space|Vector space]] | Wikipedia: [[http://en.wikipedia.org/wiki/Vector_space|Vector space]] | ||
==== Parents ==== | ==== Parents ==== | ||
- | === Requirements === | + | === Context === |
[[Vector space]], [[Set cardinality]], [[Finite sum over a monoid]] | [[Vector space]], [[Set cardinality]], [[Finite sum over a monoid]] |