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Math Help - basis theorem concept-issue !

  1. #1
    n22
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    basis theorem concept-issue !

    Hi I need some help understanding this theorem,thanks
    I dont understand this fully ..especiallly the bit about a contradiction ..what is the implication of this theorem??
    Let w be a subspace of a vector space V and dim(V) =n
    then i)dim(W)≤n;
    and
    ii)dim(W)=n iff W=V
    Proof) i) any basis B of W must contain at most n vectors ,since otherwise B is Linearly dependendent by theorem(*). Hence dim(W)≤n.
    ii)If W=V then dim(W)=dim(V)=n
    conversely suppose that dim(W)=n and let B={w,,w2,...wn} be a basis of W.
    Suppose for a contradiction that W≠V.Then there is v∈V such that v does not exist in W ..


    a0v+ a1w1+a2w2+...anWn=0
    for some scalars a0,a1,....an
    then a0=0 since otherwise..
    v=[-1/(a0)][ ]∈Span(B)=W
    so a1w1+a2w2+...anWn=0
    a1w1+a2w2+...anwn=0
    a1=a2-...=an=0
    since B is linear independent
    this shows X={v1,w1,....,wn} is linear independent,which is impossible since X contains n vectors ,so X is Linearly depenedent by theorem(*)
    ehence W=V.


    (*)theorem
    Let X={v1,v2,....Vn} be a basis of a vector space V and Y={w1,w2,...Wm} a subset of V.
    i)If m>n then Y is linear dependent..
    ii)if m<n then Y does not span V.
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  2. #2
    MHF Contributor
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    Re: basis theorem concept-issue !

    Hey n22.

    Which specific parts are you having trouble with?
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  3. #3
    n22
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    Re: basis theorem concept-issue !

    Quote Originally Posted by chiro View Post
    Hey n22.

    Which specific parts are you having trouble with?
    I dont get that bit about the contradiction ...
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  4. #4
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    Re: basis theorem concept-issue !

    Basically if two vectors are independent then it means av + bw = 0 implies that a = 0 and b = 0.

    You are assuming independence (since both sets are meant to be disjoint) and coming to a contradiction. In other words, you assume the opposite of what you are trying to prove (W = V) and arrive at a contradiction so that you can show that the initial premise (W = V) is actually true.

    If you can prove that av + bw = 0 implies that a != 0 or b != 0 then you have shown that the initial assumption (W != V) is false and that the opposite (W = V) is actually true.

    This idea known as proof by contradiction is probably the most used (and important) ways of proving things in all of mathematics.
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