Difference between revisions of "Characteristic property of the tensor product/Statement"
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** there exists a unique [[linear map]], {{M|\overline{A}:V_1\otimes\cdots\otimes V_k\rightarrow X}} such that: | ** there exists a unique [[linear map]], {{M|\overline{A}:V_1\otimes\cdots\otimes V_k\rightarrow X}} such that: | ||
*** {{M|\overline{A}\circ p\eq A}} (that is: the diagram on the right [[commutative diagram|commutes]]) | *** {{M|\overline{A}\circ p\eq A}} (that is: the diagram on the right [[commutative diagram|commutes]]) | ||
| − | Where {{M|p:V_1\times\cdots\times V_k\rightarrow V_1\otimes\cdots\otimes V_k}} by {{M|p:(v_1,\ldots,v_k)\mapsto v_1\otimes\cdots\otimes v_k}} (and is {{m|p}} is [[multilinear map|multilinear]]) {{#if:{{{full|}}}|(see '''claim 1''' for the proof of this|}} | + | Where {{M|p:V_1\times\cdots\times V_k\rightarrow V_1\otimes\cdots\otimes V_k}} by {{M|p:(v_1,\ldots,v_k)\mapsto v_1\otimes\cdots\otimes v_k}} (and is {{m|p}} is [[multilinear map|multilinear]]) {{#if:{{{full|}}}|(see '''claim 1''' for the proof of this)|}} |
<div style="clear:both;"></div><noinclude> | <div style="clear:both;"></div><noinclude> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| + | {{Theorem Of|Linear Algebra|Abstract Algebra}} | ||
</noinclude> | </noinclude> | ||
Revision as of 20:10, 3 December 2016
- Notice: this page is supposed to be transcluded, use full=true to show claims and extra things
Contents
Statement
Let [ilmath]\mathbb{F} [/ilmath] be a field and let [ilmath]\big((V_i,\mathbb{F})\big)_{i\eq 1}^k[/ilmath] be a family of finite dimensional vector spaces over [ilmath]\mathbb{F} [/ilmath]. Let [ilmath](W,\mathbb{F})[/ilmath] be another vector space over [ilmath]\mathbb{F} [/ilmath]. Then[1]:- If [ilmath]A:V_1\times\cdots\times V_k\rightarrow W[/ilmath] be any multilinear map
- there exists a unique linear map, [ilmath]\overline{A}:V_1\otimes\cdots\otimes V_k\rightarrow X[/ilmath] such that:
- [ilmath]\overline{A}\circ p\eq A[/ilmath] (that is: the diagram on the right commutes)
- there exists a unique linear map, [ilmath]\overline{A}:V_1\otimes\cdots\otimes V_k\rightarrow X[/ilmath] such that:
Where [ilmath]p:V_1\times\cdots\times V_k\rightarrow V_1\otimes\cdots\otimes V_k[/ilmath] by [ilmath]p:(v_1,\ldots,v_k)\mapsto v_1\otimes\cdots\otimes v_k[/ilmath] (and is [ilmath]p[/ilmath] is multilinear)
References
