Difference between revisions of "Tensor product of vector spaces"

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m (Alec moved page Tensor product to Tensor product of vector spaces without leaving a redirect: There's another tensor product - at 360 views ish)
 
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{{Stub page|grade=A*|msg=Demote once more of it is finished!}}
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: {{strike|Currently in the notes stage, see [[Notes:Tensor product]]}}
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__TOC__
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: {{Note|Any first-time readers should look at the [[#Abstract definition|abstract definition]] first}}
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==Definition==
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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 [[vector spaces]] over {{M|\mathbb{F} }}. Let {{M|\mathcal{F}(V_1\times\cdots\times V_k)}} denote the [[free vector space generated by|free vector space]] on {{M|\prod_{i\eq 1}^kV_k}}. We define the (abstract) ''tensor product'' of {{M|V_1,\ldots,V_k}} as{{rITSMJML}}:
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* {{M|V_1\otimes\cdots\otimes V_k:\eq\dfrac{\mathcal{F}(V_1\times\cdots\times V_k)}{\mathcal{R} } }}<!--
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NOTE ABOUT THE SIZE OF F(V_1X...XV_k)
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--><ref group="Note">Take a moment to respect just how vast the space {{M|\mathcal{F}(V_1\times\cdots\times V_k)}} is (especially if {{M|\mathbb{F}:\eq\mathbb{R} }} for example). Remember that this is ''not'' the space {{M|V_1\times\cdots\times V_k}} even though we write them as [[tuples]]. It is a ''huge'' space.
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* {{XXX|Flesh out this note}}</ref><!--
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END OF SIZEOF F(V_1X...XV_k) NOTE
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--> where {{M|\mathcal{R} }} is defined as follows:
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** {{M|\mathcal{R} }} denotes the {{link|span|vector space}} of all the union of the following two [[sets]]:
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**# {{M|\big\{ (v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)-a(v_1,\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge a\in\mathbb{F}\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\} }}
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**# {{M|\big\{(v_1,\ldots,v_{i-1},v_i+v'_i,v_{i+1},\ldots,v_k)-(v_1,\ldots,v_k)-(v_1,\ldots,v_{i-1},v'_i,v_{i+1},\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge v'_i\in V_i\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\} }}
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==Abstract definition==
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==[[Basis for the tensor product|Basis]]==
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{{:Basis for the tensor product/Statement}}
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==[[Characteristic property of the tensor product|Characteristic property]]==
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{{:Characteristic property of the tensor product/Statement}}
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
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{{Definition|Linear Algebra|Abstract Algebra}}

Latest revision as of 21:33, 22 December 2016

Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Demote once more of it is finished!
Currently in the notes stage, see Notes:Tensor product
Any first-time readers should look at the abstract definition first

Definition

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 vector spaces over [ilmath]\mathbb{F} [/ilmath]. Let [ilmath]\mathcal{F}(V_1\times\cdots\times V_k)[/ilmath] denote the free vector space on [ilmath]\prod_{i\eq 1}^kV_k[/ilmath]. We define the (abstract) tensor product of [ilmath]V_1,\ldots,V_k[/ilmath] as[1]:

  • [ilmath]V_1\otimes\cdots\otimes V_k:\eq\dfrac{\mathcal{F}(V_1\times\cdots\times V_k)}{\mathcal{R} } [/ilmath][Note 1] where [ilmath]\mathcal{R} [/ilmath] is defined as follows:
    • [ilmath]\mathcal{R} [/ilmath] denotes the span of all the union of the following two sets:
      1. [ilmath]\big\{ (v_1,\ldots,v_{i-1},av_i,v_{i+1},\ldots,v_k)-a(v_1,\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge a\in\mathbb{F}\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\} [/ilmath]
      2. [ilmath]\big\{(v_1,\ldots,v_{i-1},v_i+v'_i,v_{i+1},\ldots,v_k)-(v_1,\ldots,v_k)-(v_1,\ldots,v_{i-1},v'_i,v_{i+1},\ldots,v_k)\ \big\vert\ i\in\{1,\ldots,k\}\wedge v'_i\in V_i\wedge\forall j\in\{1,\ldots,k\}[v_j\in V_j]\big\} [/ilmath]

Abstract definition

Basis

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. Let [ilmath]n_i:\eq\text{Dim}(V_i)[/ilmath] and [ilmath]e^{(i)}_1,\ldots,e^{(i)}_{n_i} [/ilmath] denote a basis for [ilmath]V_i[/ilmath], then we claim[1]:

  • [math]\mathcal{B}:\eq\left\{e^{(1)}_{i_1}\otimes\cdots\otimes e^{(k)}_{i_k}\ \big\vert\ \forall j\in\{1,\ldots,k\}\subset\mathbb{N}[1\le i_j\le n_j]\right\} [/math]

Is a basis for the tensor product of the family of vector spaces, [ilmath]V_1\otimes\cdots\otimes V_k[/ilmath]


Note that the number of elements of [ilmath]\mathcal{B} [/ilmath], denoted [ilmath]\vert\mathcal{B}\vert[/ilmath], is [ilmath]\prod_{i\eq 1}^kn_i[/ilmath] or [ilmath]\prod_{i\eq 1}^k\text{Dim}(V_i)[/ilmath], thus:

  • [ilmath]\text{Dim}(V_1\otimes\cdots\otimes V_k)\eq\prod_{i\eq 1}^k n_i[/ilmath][1]

Characteristic property

[ilmath]\xymatrix{ V_1\times\cdots\times V_k \ar@2{->}[rr]^-A \ar@2{->}[d]_p & & W \\ V_1\otimes\cdots\otimes V_k \ar@{.>}[urr]_-{\overline{A} } }[/ilmath]
Diagram of the situation, the double-arrows is multilinear, the other is linear
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)

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)

Notes

  1. ↑ Take a moment to respect just how vast the space [ilmath]\mathcal{F}(V_1\times\cdots\times V_k)[/ilmath] is (especially if [ilmath]\mathbb{F}:\eq\mathbb{R} [/ilmath] for example). Remember that this is not the space [ilmath]V_1\times\cdots\times V_k[/ilmath] even though we write them as tuples. It is a huge space.
    • TODO: Flesh out this note

References

  1. ↑ 1.0 1.1 1.2 1.3 Introduction to Smooth Manifolds - John M. Lee
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