How to prove that a subspace $X \subset V$ ($V$ is vector space) is a proper subspace?
I've read that:
A subspace $X$ is called a proper subspace of $V$ if it is neither the whole space $V$ nor the zero subspace $\{0\}$.
But am unsure about how to display this?
If we are working with finite dimensional vector spaces (which I assume we are) then there are a few ways to do this. If $X\subseteq V$ is our vector subspace then we can simply determine what $\dim X$ is. If $0<\dim X<\dim V$ then we know that $X$ is a proper subspace. The easiest way to check this is to find a basis for the subspace and check its length.
Another more manual way to do this is to check that there is a nonzero vector in $X$, and also that $V$ contains a vector that is not in $X$.