The problem states:

Show that the only subspaces of k^2 are the vector space {0} (the zero-vector), a line, and a plane.

Now I am thinking of it as k^2=[x,y]. obviously when [x,y]=[0,0], it is the zero vector, [a,0] would be a line, and [a,b] would be a plane.

However, we are supposed to think of there being a subspace, call it S, where s is not {0} or a line. We should show that S contains two non-collinear vectors and the span of these two vectors should be k^2.

My mind is pretty much blown by this problem and any in all help would be greatly appreciated. Thanks!