Show that any subspace of a finite dimensional vector space is finite dimensional and state theorems used.
I would think that a subspace of a vector space would have fewer than or the same number of vectors in the basis as the vector space and must therefore be finite dimensional, but this does not use theorems just logic and I think my logic may be skewed. Does a subspace of a vector space have the same basis as the vector space and does this have anything to do with the proof? Can anyone help me prove this?
Any help would be appreciated. Thanks in advance.