Linear-1 6

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{u,v} is a linear independencew∈sp({u,v})Prove or disprove: {u,v,w} is a linear dependenceProof:w∈sp({u,v})⟹∃α1,α2∈F:w=α1u+α2v{u,v} is a linear independenceα1u+α2v=0⟹α1=α2=0Let β1u+β2v+β3w=0β1u+β2v+β3(α1u+α2v)=0⟹β1+β3α1=β2+β3α2=0⟹{β1=−β3α1β2=−β3α2⟹⧸⟹{β1=0β2=0β3=0For example:Let α1=1,α2=2,β3=3{β1=−β3=−3β2=−2β3=−6⟹{u,v,w} is a linear dependence

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Let V be a vector space over FLet {v1,v2},{v2,v3},{v1,v3} be linear independencesProve or disprove: {v1,v2,v3} is a linear independenceDisproof:Let V=R2Let v1=(10),v2=(01),v3=(11){v1,v2} is a linear independence{v2,v3} is a linear independence{v1,v3} is a linear independencev3=v1+v2⟹v3∈sp({v1,v2})⟹{v1,v2,v3} is a linear dependenceDisproved

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Let S={v1,v2,v3} be a linear independenceLet S∗={v1,v1+v2,v1+v2+v3}Is S∗ necessarily/always a linear independence?Solution:α1v1+α2v2+α3v3=0⟹α1=α2=α3=0β1v1+β2(v1+v2)+β3(v1+v2+v3)=0(β1+β2+β3)v1+(β2+β3)v2+β3v3⏟Linear combination of S=0⟹{β1+β2+β3=0β2+β3=0β3=0⟹{β1=0β2=0β3=0⟹S∗ is always a linear indendence

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Let V be a vector space over FU,W⊆V− vector subspaces of VLet u,v,w∈V∖{0}u∈U,u∉Ww∈Wv∈W,∄α∈F:v=αwProve: ∄α1,α2∈F:v=α1u+α2wProof:W is a vector subspace⟹∀w1,w2:w1+αw2∈WLet ∃α1,α2∈F:v=α1u+α2wLet α1=0⟹v=α2w− Contradiction!⟹α1≠0⟹u=v−α2wα1v∈W,w∈W⟹v+(−α2)w∈W⟹1α1(v+(−α2)w)∈W⟹u∈W− Contradiction!⟹∄α1,α2∈F:v=α1u+α2w

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S={(k11),(01k),(011)}Find k such that S is a linear dependenceSolution:α1⋅(k11)+α2⋅(01k)+α3⋅(011)=0⟹{α1k=0α1+α2+α3=0α1+α2k+α3=0(k00011101k10)→(k00001−k1−k0001−k0)→(k00001−k00001−k0){α1k=0α2(1−k)=0α3(1−k)=0⟹{α2=α3=0k=0α1=0k=1α1=α2=α3=0otherwisek∈{0,1}⟹S is a linear dependence

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Let V be a vector space over F{v1,v2,v3} is a linear independenceProve: {v1+v2,v2−v3,v3−2v1} is a linear independenceProof:α1v1+α2v2+α3v3=0⟹α1=α2=α3=0β1(v1+v2)+β2(v2−v3)+β3(v3−2v1)=0(β1−2β3)v1+(β1+β2)v2+(β3−β2)v3⏟Linear combination of {v1,v2,v3}=0⟹{β1−2β3=0β1+β2=0β3−β2=0(10−2011000−110)→(10−2001200−110)→(10−2001200030){β1−2β3=0β2+2β3=03β3=0⟹{β1=0β2=0β3=0⟹{v1+v2,v2−v3,v3−2v1} is a linear independence

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Let v=(ab),w=(cd)∈RProve: {v,w} is a linear dependence⟹ad−bc=0Proof:∃α1≠α2:α1v+α2w=0α1=0,α2≠0⟹α2w=0⟹c=d=0⟹ad−bc=0α1≠0,α2=0⟹α2v=0⟹a=b=0⟹ad−bc=0Let α1≠0,α2≠0⟹{aα1+cα2=0bα1+dα2=0a=0⟹c=0⟹ad−bc=0Let a≠0(ac0bd0)→(ac00d−c⋅ba0){aα1+cα2=0(d−c⋅ba)α2=0⟹d−c⋅ba=0⟹ad−bc=0

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Let V be a vector space over FS1∩S2=∅S1∪S2 is a linear independence over FProve: sp(S1∪S2)=sp(S1)⊕sp(S2)Proof:Let S1={v1,v2,…,vn}Let S2={u1,u2,…,um}S1∪S2={v1,…,vn,u1,…,um}Let s∈sp(S1∪S2)s=∑i=1nαivi⏟Linear combination of S1+∑i=1mβiui⏟Linear combination of S2⏟Linear combination of S1∪S2∑i=1nαivi∈sp(S1)∑i=1mβiui∈sp(S2)⟹s∈sp(S1)+sp(S2)⟹sp(S1∪S2)⊆sp(S1)+sp(S2)(1)Let s∈sp(S1)+sp(S2)⟹∃v∈S1,u∈S2:s=v+uv∈sp(S1)⟹v=∑i=1nαiviu∈sp(S2)⟹u=∑i=1mβiuis=v+u=∑i=1nαivi+∑i=1mβiui⏟Linear combination of S1∪S2⟹s∈sp(S1∪S2)⟹sp(S1)+sp(S2)⊆sp(S1∪S2)(2)(1)∧(2)⟹sp(S1∪S2)=sp(S1)+sp(S2)sp(S1∪S2) is a vector space⟹sp(S1)+sp(S2) is a vector space(3)Let s∈sp(S1)∩sp(S2)s∈sp(S1)⟹s=∑i=1nαivis∈sp(S2)⟹s=∑i=1mβiui⟹∑i=1nαivi=∑i=1mβiui⟹∑i=1nαivi−∑i=1mβiui⏟Linear combination of S1∪S2=0⟹α1=α2=⋯=αn=−β1=−β2=⋯=−βm=0⟹s=∑i=1nαivi=0⟹sp(S1)∩sp(S2)={0}(4)(3)∧(4)⟹sp(S1)+sp(S2)=sp(S1)⊕sp(S2)⟹sp(S1∪S2)=sp(S1)⊕sp(S2)