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Question Number 61843 by psyche last updated on 10/Jun/19

let V be a vector space and let H and K be subspace of V. show that , H+K={x:x=h+k, where h∈H and k∈K} is a subspace of V.

letVbeavectorspaceandletHandKbesubspaceofV.showthat,H+K={x:x=h+k,whereh∈Handk∈K}isasubspaceofV.

Commented by arcana last updated on 10/Jun/19

if V is a vector space over a field K. • H+K is a nonempty subset,0∈H+K • x∈H+K, x=h+k, h∈H,k∈K, where H⊆V,K⊆V ⇒x=h+k∈V. H+K⊆V •if x=h_1 +k_1 ,y=h_2 +k_2 ,h_1 ,h_2 ∈H,k_1 ,k_2 ∈K x+y=(h_1 +k_1 )+(h_2 +k_2 )=(h_1 +h_2 )+(k_1 +k_2 ), V group abelian under “+” =h+k with h=h_1 +h_2 ∈H,k=k_1 +k_2 ∈K ⇒x+y∈H+K • if a∈K,x=h+k∈H+K.a∙x=a(h+k) =a∙h+a∙k. H and K are subspaces⇒a∙h∈H,a∙k∈K ⇒a∙x∈H+K

ifVisavectorspaceoverafieldK.∙H+Kisanonemptysubset,0∈H+K∙x∈H+K,x=h+k,h∈H,k∈K,whereH⊆V,K⊆V⇒x=h+k∈V.H+K⊆V∙ifx=h1+k1,y=h2+k2,h1,h2∈H,k1,k2∈Kx+y=(h1+k1)+(h2+k2)=(h1+h2)+(k1+k2),Vgroupabelianunder‘‘+″=h+kwithh=h1+h2∈H,k=k1+k2∈K⇒x+y∈H+K∙ifa∈K,x=h+k∈H+K.a⋅x=a(h+k)=a⋅h+a⋅k.HandKaresubspaces⇒a⋅h∈H,a⋅k∈K⇒a⋅x∈H+K

Commented by arcana last updated on 10/Jun/19

define ∙: K×V→V (k,v)→k∙v ∈V

define⋅:K×V→V(k,v)→k⋅v∈V

Commented by arcana last updated on 10/Jun/19

K=S field because there is confuse with K vectorial subspace

K=SfieldbecausethereisconfusewithKvectorialsubspace

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