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Math Help - find a basis for span(T) as a subspace of R3

  1. #1
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    find a basis for span(T) as a subspace of R3

    what if i have

    T = { [-2,1,3] , [4,-1,2] , [2,0,5] }

    if I form matrix A with its columns as the vectors above, the RREF of A has a row of zeros

    furthermore, det(A) = 0,

    how could i find the basis?
    or what should be my conclusion?
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  2. #2
    MHF Contributor

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    The fact that the RREF has a row of zeros tells you that the three vectors are not independent- the dimension of the subspace is less than 3. But it is easy to see that the first two are not multiples of one another so they are independent- the dimension of the subspace is at least 2. Those two together tell us that the dimension of the subspace is exactly two and any two independent vectors, the first two of these, for example, form a basis.

    (Actually, since none of these given vectors is a multiple of any one of the others, any two of them form a basis.)
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