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Surjectivity and linear maps question

  1. May 22, 2006 #1
    In my head this proof seems obvious, but I am unable to write it rigorously. :cry: Any help would be appreciated!

    Prove that it T is a linear map from F^4 to F^2 such that

    kernel T ={(x1, x2, x3, x4) belonging to F^4 | x1 = 5x2 and x3 = 7x4}, then T is surjective.
     
  2. jcsd
  3. May 22, 2006 #2

    matt grime

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    What's the dimension of the image? (think in terms of the dimension of the domain and the dimension of the kernel)
     
  4. May 22, 2006 #3
    the dimension of the kernel is two, the dimension of the range is two

    so F^2 equals dim range and therefore is surjective?
     
  5. May 22, 2006 #4
    dimension of kernel equal 2. Dimension of range equals 2.

    dimension of domain equals 4. Since dim range = 2 and F^2 is the whole space of the range, then it is surjective?
     
  6. May 22, 2006 #5

    matt grime

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    F^2 is a vector space, so it can't equal a dimension.

    The image is a 2-d subspace of a 2-d space, so it is all of it.
     
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