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Math Help - vector space

  1. #1
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    vector space

    how do you show that for every set S, S ⊂ span S?


    my working:

    let S be the set of real numbers in n dimension.

    then S = { x1,x2,...,xn l x1,x2,...,xn are real numbers}

    then span S = a1x1+ a2x2+...+anxn

    for all xi ∈ S, where 1<i<n

    xi = a1x1+ a2x2+.+aixi+..+anxn iff all the coefficients are 0 except ai=1

    thus xi ⊂ a1x1+ a2x2+.+aixi+..+anxn

    can i prove this question like this? how should i define S since the proof requires that the statement is true for all S?

    thanks
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  2. #2
    A Plied Mathematician
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    So, \text{span}(S) is the set of all linear combinations of vectors in S. Correct? Let s\in S. Then s=1\cdot s\in\text{span}(S), because it's a linear combination of vectors in S.

    Doesn't this do it?
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  3. #3
    Member HappyJoe's Avatar
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    I don't quite understand your proof.

    But to show the statement for a general set S, notice that span S consists of all finite linear combinations of elements from S. In particular, take some element x in S. Then x=1x is a finite linear combination involving elements of S, and hence x is in span S.
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  4. #4
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    thanks!! i was thinking that i need to define the vectors in S and then define the span of S.
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  5. #5
    A Plied Mathematician
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    You're very welcome for my contribution, whatever that was. Have a good one!
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