Linear-1 5

Linear 1 5

A,B∈Fn×nAB=I⟺BA=I∃A−1,B−1⟺∃(AB)−1
U+W={u+w∣u∈U,w∈W}U,W⊆VU⊆U+W(∀u:u=u+0W)W⊆U+W(∀w:w=0U+w)U+W⊆V(∀u,w:u+w=v1+v2∈V)1.0V∈U+W0V=0U+0W∈U+W2.v,v~∈U+Wv∈U+W⟹v=u+wv~∈U~+W~⟹v~=u~+w~v+αv~=u+w+α(u~+w~)==u+αu~+w+αw~=u+u~+w+w~
U={(x0)∣x∈R}W={(0x)∣x∈R}U+W={(xx)∣x∈R}=R2{(xy0)}+{(0xy)}=R3

Direct sum #definition

U,W⊆V over FU⊕W−direct sum of U and W(∀v:∃u∈U,w∈W:v=u+w)∧U∩W={0}⟺U⊕W=V

Uniqueness of vector spaces sum #lemma

U,W⊆V over FU⊕W=V⟺∀v∈V:∃!u∈U,w∈W:v=u+wProof:1.Let U⊕W=V∃u∈U,w∈W:v=u+wLet u~∈U,w~∈W:v=u~+w~⟹u+w=u~+w~⟹u~−u=w−w~⟹u~−u∈U∧u~−u∈W⟹u~−u∈U∩W⟹u~−u∈{0}⟹u~−u=0⟹u=u~⟹∃!u∈U,w∈W:v=u+w2.Let∃!u∈U,w∈W:v=u+w∃!u∈U,w∈W:v=u+w⟹∃u∈U,w∈W:v=u+w⟺U+W=V{0}⊆U∩WLet v∈U∩Wv=v+0,v=0+v∃!u∈U,w∈W:v=u+w⟹0+v=v+0⟹v=0⟹U∩W⊆{0}⟹U∩W={0}
S={(01),(11)}(35)=−2(01)+5(11)∃S1,S2,S3,…,Sn∈S∃α1,α2,α3,…,αn∈Fv=α1S1+α2S2+⋯+αnSnS={1,2+x,1+x2}2+x−x2=α(1)+β(2+x)+γ(1+x2)α+2β+βx+γ+γx2=(α+2β+γ)1+(β)x+(γ)x2{α+2β+γ=2β=1γ=−1α=1,β=1,γ=−1

Span #definition

V over F,S⊆Vsp(S) (span S)sp(S)={α1s1+α2s2+⋯+αnsn|n∈Nα1,α2,…,αn∈Fs1,s2,…,sn∈S}

Span cannot exceed vector space #lemma

S⊆V⟹sp(S) is a subspace of VProof:1.S=∅S=∅⟹sp(S)={0}⟹sp(S) is a subspace of V2.S≠∅2.10→∈sp(S)S≠∅⟹∃s∈S:0⋅s∈sp(S)⟺0→∈sp(S)2.2Let u,v∈sp(S)⟹u=α1S1+…αnSn⟹v=β1S1+⋯+βnSn+0(S~1+S~2+…)⟹u+αv=(α1+αβ1)S1+…(αn+αβn)Sn∈sp(S)