# Thread: Find the subspaces and show a linear transformation permutes subspace

1. ## 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.

2. 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!