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]

• The alternative notation [math]\bigoplus_{i\in K}^\text{ext}[/math] is sometimes used

[math]V=\bigoplus\mathcal{F}[/math] or [math]V=\bigoplus_{i\in I}[/math] where the following hold:

1. [math]V=\sum_{i\in I}V_i[/math] - that is that [ilmath]V[/ilmath] is the sum (or join) of the family [ilmath]\mathcal{F} [/ilmath]
2. [math]\forall i\in I[/math] we have [math]V_i\cap\left(\sum_{j\ne i}V_j\right)=\{0\}[/math]

• For the second condition each [ilmath]V_j[/ilmath] is called a direct summand of [ilmath]V[/ilmath]
• If [ilmath]\mathcal{F} [/ilmath] is finite, that is [math]\mathcal{F}=\{V_1,\cdots,V_n\}[/math] then we often write:

  [math]V=V_1\oplus\cdots\oplus V_n[/math]

• If [ilmath]V=S\oplus T[/ilmath] then we call [ilmath]T[/ilmath] a complement of [ilmath]S[/ilmath] in [ilmath]V[/ilmath]
• The [ilmath]2^\text{nd} [/ilmath] condition is stronger than saying the members of [ilmath]\mathcal{F} [/ilmath] are pairwise disjoint - the book makes this clear although I see it as obvious. (Even though they're not quite pairwise disjoint!)