Notes:Vector space operations

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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 [math]\forall s\in S[/math] there are unique [math]u\in U[/math] and [math]w\in W[/math] such that [math]s=u+w[/math]

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


  • 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 [math]\oplus[/math] 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 [math]\prod[/math] [math]\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\}[/math] Will check with isomorphism later
External Direct Sum [math]\bigoplus^\text{ext}_{i\in K}[/math] [math]\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\}[/math]
NOTICE: direct product and external direct sum are the same for finite family!
Internal direct sum [math]\bigoplus[/math] [math]V=\sum_{i\in K}S_i[/math] (for subspaces [math]S_i[/math]) and [math]\forall i\in I\ S_i\cap\left(\sum_{j\ne i}S_j\right)=\{0\}[/math]


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