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2 vector proofs

  1. Aug 15, 2009 #1
    1
    Suppose S is a set of n linearly independet in the n-dimensional vectorspace V. Prove that S is a basis for V.

    My try at this proof is:
    For S to be a basis for V it has to span V and the vectors in S needs to be linearly independent. But they have allreade sad that the vectors in S are linearly independent, so we only needs to show that it spans V.

    But since V is n-dimensional it means that n linearly independent vectors in V span it, hence since S has n linearly independent vectors and is in V, S is a basis for V? Is this right?

    2
    Suppose that S is a set if n vectors that span the n-dimensional vector space V. Prove that S is a basis for V.

    Now we need to show that the vectors in S is lineraly independent, right? But since n is n-dimensional it means that n lineraly independent vectors span it, since S is a set of vectors that spans V and S has n vectors, S is lineraly independent? Have I proved this one right?
     
  2. jcsd
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