Difference between revisions of "Addition of vector spaces"

[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]

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]

[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

[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]

[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]