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Condition for equality between subspaces. 
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#1
Dec312, 12:12 PM

P: 818

Hi,
1. The problem statement, all variables and given/known data What would be the/a condition on vectors in K so that V=W, where V is a vector space which K={v1,v2,v3,v4} spans, and W is a subspace of V defined thus: W=Sp{v1+v2,v2+v3,v3+v4,v4+v1} 2. Relevant equations 3. The attempt at a solution I believe V would be equal to W if W were linearly independent, but by writing that mathematically I get a condition for the scalars, not the vectors in K themselves. I hope one of you could assist. Thanks in advance! 


#2
Dec312, 12:17 PM

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P: 18,086

[tex]\{v_1+v_2,v_2+v_3,v_3+v_4,v_4+v_1\}[/tex] are linearly independent. That would indeed be the correct condition. 


#3
Dec312, 02:04 PM

P: 818

I have tried to find conditions so that:
a1v1 + a2v2 + a3v3 + a4v4 = v1(b1+b4) + v2(b2+b1) + v3(b3+b2) + v4(b4+b3). But that yielded conditions on the scalars, not the vectors. Can conditions on the vectors themselves be found? 


#4
Dec312, 02:10 PM

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P: 18,086

Condition for equality between subspaces.
[itex]\alpha(v_1+v_2)+\beta(v_2+v_3)+\gamma (v_3+v_4)+\delta(v_4+v_1)=0[/itex] only if [itex]\alpha=\beta=\gamma=\delta=0[/itex]. Now, try to use that [itex]\{v_1,v_2,v_3,v_4\}[/itex] is a basis. 


#5
Dec312, 04:03 PM

P: 818

These yielded alpha=delta=beta=gamma.
But how does this affect the vectors in K themselves? I mean, what is then the condition on v1,v2,v3,v4? 


#6
Dec312, 05:05 PM

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P: 18,086

[tex]\alpha(v_1+v_2)+\beta(v_2+v_3)+\gamma(v_3+v_4)+ \delta(v_4+v_1)=0[/tex] and if you substitute [itex]\alpha[/itex] for [itex]\gamma[/itex] and [itex]\alpha[/itex] for [itex]\delta[/itex] and [itex]\beta[/itex]? 


#7
Dec312, 06:42 PM

P: 818

You get alpha*0=0. How does that help?



#8
Dec312, 07:04 PM

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#9
Dec312, 07:30 PM

P: 818

Let us go back a bit, momentarily.
I am slightly confused. Why is it that for V to be equal to W, the elements in W must be linearly independent? Is it because dimV is equal to or less than the number of elements in K, i.e. 4? Furthermore, I know that if the elements in K are linearly independent, then V is not equal to W. Does that mean that for any K whose elements are linearly dependent, V would be equal to W? 


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