Difference between revisions of "Addition of vector spaces"

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

Operations: (given $u_i,v_i\in V_i$ and $c$ is a scalar in $F$)

• $$(u_1,\cdots,u_n)+(v_1,\cdots,v_n)=(u_1+v_1,\cdots,u_n+v_n)$$
• $$c(v_1,\cdots,v_n)=(cv_1,\cdots,cv_n)$$

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

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

Are isomorphic

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

• The alternative notation $$\bigoplus_{i\in K}^\text{ext}$$ is sometimes used

$$V=\bigoplus\mathcal{F}$$ or $$V=\bigoplus_{i\in I}$$ where the following hold:

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

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

  $$V=V_1\oplus\cdots\oplus V_n$$

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