Difference between revisions of "Addition of vector spaces"

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(Created page with " ==Definitions== {| class="wikitable" border="1" |- ! Name ! Expression ! Notes |- !colspan="3" | Finite |- |rowspan="3" | External direct sum | Given <math>V_1,\cdots,V_n...")
 
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<math>V_1+V_2+\cdots+V_n</math>
 
<math>V_1+V_2+\cdots+V_n</math>
 
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!colspan="3" | For any family of vectors (here {{M|K}} will denote an [[Indexing set|indexing set]] and <math>\mathcal{F}=\left\{V_i|i\in K\right\}</math> (a family of [[Vector space|vector spaces]] over {{M|F}}))
 
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| [[Direct product]]
 
| [[Direct product]]
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| Given <math>\mathcal{F}=\left\{V_i|i\in K\right\}</math> (a family of [[Vector space|vector spaces]] over {{M|F}})<br/>
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| <math>V=\prod_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\in K\right\}</math>
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<math>V=\prod_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\in K\right\}</math>
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| Generalisation of the external direct sum
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|-
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|rowspan="3" | [[External direct sum]]
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| <math>V=\mathop{\boxplus}_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\in K,\ f\text{ has finite support}\right\}</math>
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| '''Note:''' <br/>
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* The alternative notation <math>\bigoplus_{i\in K}^\text{ext}</math> is sometimes used
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!colspan="2" | Finite support:
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|-
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| A function {{M|f}} has finite support if {{M|1=f(i)=0}} for all but finitely many {{M|i\in K}}
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| So it is "zero almost everywhere" - the set <math>\{f(i)|f(i)\ne 0\}</math> is finite.
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|-
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| [[Internal direct sum]]
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| <math>V=\bigoplus\mathcal{F}</math> or <math>V=\bigoplus_{i\in I}</math> where the following hold:
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# A
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# B
 
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Revision as of 10:41, 24 April 2015

Definitions

Name Expression Notes
Finite
External direct sum Given [math]V_1,\cdots,V_n[/math] which are vector spaces over the same field [ilmath]F[/ilmath]:

[math]V=\mathop{\boxplus}^n_{i=1}V_i=\left\{(v_1,\cdots,v_n)|v_i\in V_i,\ i=1,2,\cdots,n\right\}[/math]
Often written: [math]V=V_1\boxplus V_2\boxplus\cdots\boxplus V_n[/math]

This is the easiest definition, for example [math]\mathbb{R}^n=\mathop{\boxplus}^n_{i=1}\mathbb{R}=\underbrace{\mathbb{R}\boxplus\cdots\boxplus\mathbb{R}}_{n\text{ times}}[/math]

Operations: (given [ilmath]u,v\in V[/ilmath] where [ilmath]u_i[/ilmath] and [ilmath]c[/ilmath] is a scalar in [ilmath]F[/ilmath])

  • [math](u_1,\cdots,u_n)+(v_1,\cdots,v_n)=(u_1+v_1,\cdots,u_n+v_n)[/math]
  • [math]c(v_1,\cdots,v_n)=(cv_1,\cdots,cv_n)[/math]
Alternative form
[math]V=\mathop{\boxplus}^n_{i=1}V_i=\left\{\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right|f(i)\in V_i\ \forall i\in\{1,\cdots,n\}\right\}[/math] Consider the association:

[math](v_1,\cdots,v_n)\mapsto\left[\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right|f(i)=v_i\ \forall i\right][/math]
That is, that maps a vector to a function which takes a number from 1 to [ilmath]n[/ilmath] to the [ilmath]i^\text{th} [/ilmath] component, and:
Given a function [math]f:\{1,\cdots,n\}\rightarrow\cup_{i=1}^nV_i[/math] where [math]f(i)\in V_i\ \forall i[/math] we can define the following association:
[math]f\mapsto(f(1),\cdots,f(n))[/math]
Thus:

  • [math]V=\mathop{\boxplus}^n_{i=1}V_i=\left\{\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right|f(i)\in V_i\ \forall i\right\}[/math]
  • [math]V=\mathop{\boxplus}^n_{i=1}V_i=\left\{(v_1,\cdots,v_n)|v_i\in V_i,\ \forall i\right\}[/math]

Are isomorphic

Sum of vector spaces Given [ilmath]V_1,\cdots,V_n[/ilmath] which are vector subspaces of [ilmath]V[/ilmath]

[math]\sum^n_{i=1}V_i=\left\{v_1+\cdots+v_n|v_i\in V_i,\ i=1,2,\cdots,n\right\}[/math]
Sometimes this is written: [math]V_1+V_2+\cdots+V_n[/math]

For any family of vectors (here [ilmath]K[/ilmath] will denote an indexing set and [math]\mathcal{F}=\left\{V_i|i\in K\right\}[/math] (a family of vector spaces over [ilmath]F[/ilmath]))
Direct product [math]V=\prod_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\in K\right\}[/math] Generalisation of the external direct sum
External direct sum [math]V=\mathop{\boxplus}_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\in K,\ f\text{ has finite support}\right\}[/math] Note:
  • The alternative notation [math]\bigoplus_{i\in K}^\text{ext}[/math] is sometimes used
Finite support:
A function [ilmath]f[/ilmath] has finite support if [ilmath]f(i)=0[/ilmath] for all but finitely many [ilmath]i\in K[/ilmath] So it is "zero almost everywhere" - the set [math]\{f(i)|f(i)\ne 0\}[/math] is finite.
Internal direct sum [math]V=\bigoplus\mathcal{F}[/math] or [math]V=\bigoplus_{i\in I}[/math] where the following hold:
  1. A
  2. B

References

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