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Math Help - sequence and basis of subspace!! help!

  1. #1
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    sequence and basis of subspace!! help!

    In the real vector space R^4, find a basis for the subspace Span(S), where S is
    the sequence
    (1, 0, 2, 0), (2, 2,−2, 4), (1, 1,−1, 2), (0, 2,−6, 4), (2, 0, 4, 0).

    The solution which i downloaded from my uni's web is (1,0,2,0)
    (0 ,1 ,-3 ,2).
    this method i can see they wrote the vectors in rows and do suitable row operations.
    But i used the method which taught in the lecture ,and i got the basis is (1,0,2,0),(1,1,-2,2),(0,2,-6,4).
    (i wrote the vectors in columns and did row operations)

    which one is right? can the subspace Span(S) have different basis??? Thank you very much!!
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  2. #2
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    First of all, obviously there are inifinite possibilities for bases in every vector space. You can always multiply all of a given basis elements in a scalar and get a new, different, legitimate basis, for example.
    However, the dimension of a vector space is defined well. consequently, you and your source cannot both be right, since you found 3 linearly independent vectors which form (in your opinion) a basis, meaning that dim(V) = 3, while according to your source, dim(V) = 2.
    Sorry to say that your source is indeed correct. The vector space is spanned by these two vectors: {(1,0,2,0),(0,1,-3,2)}.
    That means that each vector, including the others you've written, can be given as a linear combination of these two vectors.
    To find that out - write all of the vectors as columns (or rows) in a matrix, and use Gaussian elimination to find the two linear independent vectors I've mentioned.
    (Notice that there are other possibilities, but these two are the ones I got as well by elementry actions, and I think so should you.)
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  3. #3
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    Im clear with this now. i figured out the mistake i made, because i copied one wrong vector from the question. but anyway thank you very much.
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