# Notes:Vector space operations

This page is a notes page for evidence of operations on vector spaces - that is the definitions of these operations according to different authors

## Serge Lang - Linear Algebra - UTM

Name Symbol Definition Comment
Sum [ilmath]+[/ilmath] Given [ilmath]U[/ilmath], [ilmath]W[/ilmath] subspaces of [ilmath]V[/ilmath] we define the sum of [ilmath]U+V[/ilmath] to be the subset of [ilmath]V[/ilmath] consisting of all sums [ilmath]u+w\vert u\in U\wedge v\in V[/ilmath] looks internal
Direct Sum [ilmath]\oplus[/ilmath], [ilmath]\bigoplus^n_{i=1}[/ilmath] Quoting: We say [ilmath]S[/ilmath] is the direct sum of [ilmath]U[/ilmath] and [ilmath]W[/ilmath] if $\forall s\in S$ there are unique $u\in U$ and $w\in W$ such that $s=u+w$

Leads to theorem: if [ilmath]U\cap W=\{0\}[/ilmath] then [ilmath]U+W[/ilmath] is a direct sum

looks internal

CAREFUL WHEN GENERALISING (see Roman - Internal Direct Sum)

Direct Product [ilmath]\times[/ilmath], [ilmath]\prod^n_{i=1}[/ilmath] Vector space on tuples of vectors Looks like Cartesian product and external direct sum

Violates:

• Direct sum [ilmath]\implies[/ilmath] external when internal or external isn't mentioned. However subspace is mentioned (which means we can do internal)

## Linear Algebra via Exterior Products - Sergei Winitzki

Name Symbol Definition Comment
Direct Sum $\oplus$ Given two vector spaces over the same field, we define a new one as the vector space of tuples of vectors from each - Direct Product to Lang Contradicts Lang Linear Algebra - looks external

## Advanced Linear Algebra - Steven Roman

Name Symbol Definition Comment
Sum [ilmath]+[/ilmath], [ilmath]\sum[/ilmath] All finite sums from the union of the family of subspaces (inline with Lang's sum) Inline with Lang
External direct sum [ilmath]\boxplus[/ilmath] (finite) Tuples from the cartesian product of the family (for finite) Lang calls Direct Product, Winitzki calls direct sum
Direct product $\prod$ $\prod_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\right\}$ Will check with isomorphism later
External Direct Sum $\bigoplus^\text{ext}_{i\in K}$ $\bigoplus^\text{ext}_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\wedge f\text{ has finite support}\right\}$
NOTICE: direct product and external direct sum are the same for finite family!
Internal direct sum $\bigoplus$ $V=\sum_{i\in K}S_i$ (for subspaces $S_i$) and $\forall i\in I\ S_i\cap\left(\sum_{j\ne i}S_j\right)=\{0\}$

## Thoughts

Analogies with the box and product topologies of infinite things comes to mind.