View source for Addition of vector spaces


==Definitions==
{| class="wikitable" border="1"
|-
! Name
! Expression
! Notes
|-
!colspan="3" | Finite
|-
|rowspan="3" | [[External direct sum]]
| Given <math>V_1,\cdots,V_n</math> which are [[Vector space|vector spaces]] over the same field {{M|F}}:<br/>
<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><br/>
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><br/>
'''Operations:''' (given {{M|u,v\in V}} where {{M|u_i}} and {{M|c}} is a scalar in {{M|F}})
* <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>
|-
!colspan="2" | 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:<br/>
<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><br/>
That is, that maps a vector to a function which takes a number from 1 to {{M|n}} to the {{M|i^\text{th} }} component, and:<br/>
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:<br/>
<math>f\mapsto(f(1),\cdots,f(n))</math><br/>
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|isomorphic]]
|-
| [[Sum of vector spaces]]
| Given {{M|V_1,\cdots,V_n}} which are [[Vector space|vector subspaces]] of {{M|V}}<br/>
<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><br/>
Sometimes this is written:
<math>V_1+V_2+\cdots+V_n</math>
|
|-
| [[Direct product]]
| Given <math>\mathcal{F}=\left\{V_i|i\in K\right\}</math> (a family of [[Vector space|vector spaces]] over {{M|F}})<br/>
<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>
|}

==References==
<references/>

{{Definition|Linear Algebra}}