Difference between revisions of "Addition of vector spaces"

Latest revision as of 18:02, 18 March 2016

Notes

See Notes:Vector space operations

Definitions

All of this comes from the same reference^[1]

Name	Expression	Notes
Finite
External direct sum	Given which are vector spaces over the same field $F$ : $V=\mathop{\boxplus}^n_{i=1}V_i=\left\{(v_1,\cdots,v_n)\|v_i\in V_i,\ i=1,2,\cdots,n\right\}$ Often written: $V=V_1\boxplus V_2\boxplus\cdots\boxplus V_n$	This is the easiest definition, for example $\mathbb{R}^n=\mathop{\boxplus}^n_{i=1}\mathbb{R}=\underbrace{\mathbb{R}\boxplus\cdots\boxplus\mathbb{R}}_{n\text{ times}}$ Operations: (given $u_i,v_i\in V_i$ and $c$ is a scalar in $F$ ) $(u_1,\cdots,u_n)+(v_1,\cdots,v_n)=(u_1+v_1,\cdots,u_n+v_n)$ $c(v_1,\cdots,v_n)=(cv_1,\cdots,cv_n)$
	Alternative form
	$V=\mathop{\boxplus}^n_{i=1}V_i=\left\{\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right\|f(i)\in V_i\ \forall i\in\{1,\cdots,n\}\right\}$	Consider the association: $(v_1,\cdots,v_n)\mapsto\left[\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right\|f(i)=v_i\ \forall i\right]$ 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 of vector spaces	Given $V_1,\cdots,V_n$ which are vector subspaces of $V$ $\sum^n_{i=1}V_i=\left\{v_1+\cdots+v_n\|v_i\in V_i,\ i=1,2,\cdots,n\right\}$ Sometimes this is written: $V_1+V_2+\cdots+V_n$
For any family of vectors (here $K$ will denote an indexing set and (a family of vector spaces over $F$ ))
Direct product	$V=\prod_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right\|f(i)\in V_i\ \forall i\in K\right\}$	Generalisation of the external direct sum
External direct sum	$V=\mathop{\boxplus}_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right\|f(i)\in V_i\ \forall i\in K,\ f\text{ has finite support}\right\}$	Note: The alternative notation $\bigoplus_{i\in K}^\text{ext}$ is sometimes used
	Finite support:
	A function $f$ has finite support if $f(i)=0$ for all but finitely many $i\in K$	So it is "zero almost everywhere" - the set $\{f(i)\|f(i)\ne 0\}$ is finite.
Internal direct sum	Given a family of subspaces of $(V,F)$ , , the internal direct sum is defined as follows: $V=\bigoplus\mathcal{F}$ or $V=\bigoplus_{i\in I}$ where the following hold: - that is that $V$ is the sum (or join) of the family $\mathcal{F}$ $\forall i\in I$ 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!)

References

Jump up ↑ Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics

[1] Jump up ↑ Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics

[1]

@@ Line 1: / Line 1: @@
+==Notes==
+* See [[Notes:Vector space operations]]
 ==Definitions==
+All of this comes from the same reference<ref>Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics</ref>
 {| class="wikitable" border="1"
 |-
@@ Line 14: / Line 18: @@
 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}})
+'''Operations:''' (given {{M|u_i,v_i\in V_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>
@@ Line 55: / Line 59: @@
 |-
 | [[Internal direct sum]]
-| <math>V=\bigoplus\mathcal{F}</math> or <math>V=\bigoplus_{i\in I}</math> where the following hold:
+| Given a family of subspaces of {{M|(V,F)}}, <math>\mathcal{F}=\{V_i|i\in I\}</math>, the internal direct sum is defined as follows:<br/>
-# A
+<math>V=\bigoplus\mathcal{F}</math> or <math>V=\bigoplus_{i\in I}</math> where the following hold:
-# B
+# <math>V=\sum_{i\in I}V_i</math> - that is that {{M|V}} is the sum (or join) of the family {{M|\mathcal{F} }}
+# <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 {{M|V_j}} is called a ''direct summand'' of {{M|V}}
+* If {{M|\mathcal{F} }} 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 {{M|1=V=S\oplus T}} then we call {{M|T}} a '''complement of {{M|S}} in {{M|V}}'''
+* The {{M|2^\text{nd} }} condition is stronger than saying the members of {{M|\mathcal{F} }} are pairwise disjoint - the book makes this clear although I see it as obvious. (Even though they're not quite pairwise disjoint!)
 |}

Difference between revisions of "Addition of vector spaces"

Latest revision as of 18:02, 18 March 2016

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