Difference between revisions of "Characteristic property of the tensor product/Statement"
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-->Let {{M|\mathbb{F} }} be a [[field]] and let {{M|\big((V_i,\mathbb{F})\big)_{i\eq 1}^k}} be a family of ''[[dimension (vector space)|finite dimensional]]'' [[vector spaces]] over {{M|\mathbb{F} }}. Let {{M|(W,\mathbb{F})}} be another vector space over {{M|\mathbb{F} }}. Then{{rITSMJML}}: | -->Let {{M|\mathbb{F} }} be a [[field]] and let {{M|\big((V_i,\mathbb{F})\big)_{i\eq 1}^k}} be a family of ''[[dimension (vector space)|finite dimensional]]'' [[vector spaces]] over {{M|\mathbb{F} }}. Let {{M|(W,\mathbb{F})}} be another vector space over {{M|\mathbb{F} }}. Then{{rITSMJML}}: | ||
| − | * If {{M|A:V_1\times\cdots\times V_k\rightarrow W}} | + | * If {{M|A:V_1\times\cdots\times V_k\rightarrow W}} is any [[multilinear map]] |
** 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> | ||
Latest revision as of 22:07, 20 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] is 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
