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Math Help - Find the subspaces and show a linear transformation permutes subspace

  1. #1
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    Find the subspaces and show a linear transformation permutes subspace

    Let V = K^2 be a vector space over a field finite field K, in which |K| = 4. Show that V has exactly five 1-dimensional subspaces, say U_1,...,U_5 and that a non-singular linear transformation of V permutes these five subspaces.
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  2. #2
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    Also, the question goes on to "associate to a the map \sigma_a\ in S_5 (where S_5 is the symmetric group of 5 elements under composition).

    This seems very unclear to me - what does it actually mean? I've got to show that the map \sigma: a \rightarrow \sigma_a defines a monomorphism from SL(2,4) into S_5 which I think would be manageable provided I understood what was meant by the "corresponding permutation" in the first place!
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