Are all subspaces of equal dimension (of a vector space) the same? I haven't quite gotten my head around dimension, bases, and subspaces. It seems intuitively true, but are all subspaces of equal dimension of the same vector space the same?
If so, does it follow from the definitions of dimension, subspace, and vector space, or does it need to be proven?
Thanks

Here are some more intuitive "definitions".
Dimension - Number of degrees of freedom of movement. One-dimensional implies only one direction of movement: up and down a line. Two-dimensional means two distinct directions of movement, spanning a plane, etc.
Basis - The distinct directions of movement. One-dimensional movement only says we are moving along a line, but it does not specify which line. The basis tells us the directions we can move in. 
Can you see why the dimension of a subspace is not enough information to uniquely identify it? Why do we need to know the basis as well?
