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vector_space_basis [2013/09/07 19:31] nikolaj |
vector_space_basis [2014/12/02 16:37] nikolaj |
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| ===== 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: $n\equiv\left|(B'\right|$ | |
| | $v_1,\dots,v_n\in B'$ | | | $v_1,\dots,v_n\in B'$ | | ||
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| | $x\in V$ | | | $x\in V$ | | ||
| - | | @#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. | ||
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| 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]] | ||
