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Math Help - Linear Algebra Question

  1. #1
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    Linear Algebra Question

    Hi , I need clarification about something

    Let's say V is a Vector space over the field F

    A is a subspace of V, A={v1,v2,......,vn}

    So my question is what is the difference between Span(A) and the subspace A? Isn't it kinda same thing? Span(A) is the linear combination of the vectors of A with the product of arbitrary scalars.

    And subspace has axioms such as it is closed under addition and multiplication with scalar, so if there is a vector u that belongs to the Span(A) he belongs to the subspace A as well because A is closed under addition right?

    So my questions is is it possible to have a vector u that belongs to Span(A) but not the subspace A={v1,v2,v3,.......,vn}?
    I hope my question was clear, thanks!
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  2. #2
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    Re: Linear Algebra Question

    I forgot to ask one more question that I had on mind,
    If u belongs to Span(A) does it mean that u is equal to vi when 1<i<k (including 1 and k) that Span(v1,v2,.....,vi,...vn)
    or does it mean that with the linear combinations of vectors of A I can get the vector u?

    In other words doest it mean u belongs to the set A or you can get u by the linear combinations of the vectors that belong to A
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  3. #3
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    Re: Linear Algebra Question

    If A is a subspace, it is spanned by any set of linearly independent vectors in A, or any set of vectors in A which contain the max number of linearly independent vectors; but that includes A itself, so I suppose you could say spanA = A, which I didn't see when I started this reply. Interesting. Thanks.
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  4. #4
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    Re: Linear Algebra Question

    Typically we talk about "span(A)" for some set of vectors, not, in general, a subspace. Span(A) can be defined as "the smallest vector space containing A". If A is itself a vector space, then it follows immediately that span(A)= A.
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  5. #5
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    Re: Linear Algebra Question

    By definition, given a finite set of vectors S, the span of S (spanS) is the vector space consisting of all linear combinations of the vectors of S.

    Since A consists of an infinite set of vectors, spanA is undefined, contrary to my original speculation, ie, spanA is not A.
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  6. #6
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    Re: Linear Algebra Question

    But since S is closed under addition and multiplication with scalars, Span(S) has to be equal to S, isn't it ? Because linear combinations of the vectors of S are still in the subspace S
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  7. #7
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    Re: Linear Algebra Question

    S is a finite set of vectors. Span of S is an infinite set of vectors.

    Let S = (0,1) and (1,0). Then span of S consists of all (x,y) and so is an infinite set.
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