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Linear independence and span

  1. Dec 4, 2005 #1
    Is there a linear algebra theorem or fact that says something like

    For a linear transformation T:Rn -> Rm and its standard m x n matrix A:
    (a) If the columns of A span Rn the transformation is onto.
    (b) If the columns of A are linearly independent the transformation is one-to-one.

    Is this correct? I can't find it anywhere in my textbook but it may have been mentioned in lecture. Any insight would be appreciated.
  2. jcsd
  3. Dec 5, 2005 #2


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    If m>n, then can T be onto? If the colomns are independent, T(Rn) will be an n-dimensional subspace of Rm.

    b) is true, as you can easily prove yourself. Show that T(v1)=T(v2) implies that v1=v2.
    Hint: If you column-vectors are independent you can use them as a basis.
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