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Math Help - Finding a Basis from Set of Vectors

  1. #1
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    Finding a Basis from Set of Vectors

    I have the following problem along with my solution.













    My textbook lists the correct answer as



    Is my solution correct? So, once I arrive at the reduced-row echelon form of the matrix A, I can tell which vectors in the set are linearly independent by the "leading ones" in my matrix A. Those vectors will always form a basis?

    I have two vectors in R^4 and my professor's answer was two vectors in R^4. The book lists two vectors in R^2. Is it possible to form a basis for the subspace that has some combination of vectors in R, R^3, R^5, R^n?


    My professor transposes the matrix A and then proceeds to find a basis. He arrives at a different basis. There are infinitely many basis correct? My professor said that he transposes A so that he can use row operations and not change the solution. Well, if there are infinitely many basis, why does this matter? Could someone please clarify this problem for me?

    Thanks
    Last edited by Fourier; October 23rd 2007 at 06:32 PM.
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  2. #2
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    Opalg's Avatar
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    If W is a subspace of R^3 then any basis of W must consist of vectors in R^3. The correct procedure is to write the given vectors in W as the rows of a matrix and then reduce it to echelon form. In other words, the prof is correct to use the transpose of A. In the row-reduced form of the matrix, the nonzero rows form a basis for W.

    A subspace W of R^n has infinitely many bases, but they must all consist of vectors in W, and therefore in R^n. You can never have basis vectors in a different-dimensional space from the one you started out in.
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  3. #3
    MHF Contributor kalagota's Avatar
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    hmm, actually, you may use the definition of a basis.
    you are already given that the set is a spanning set, therefore, you just need to looke if they are linearly independent.
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