I am having trouble grasping an intuitive understanding of why a subspace needs to have the zero vector. I understand to satisfy the axioms the zero vector is needed, like: "a vector $v$ is an element of a subspace then $-v$ must be also and $v+(-v) = 0$" but I am not having trouble understanding the properties of a subspace, instead I am having trouble grasping an intuitive understanding of the zero vector. For example why would a line that does not pass through the origin not be considered a subspace of $\Bbb R^n$? I know that line would not satisfy the zero vector but wouldn't that line still satisfy addition and scalar axioms? Can anyone provide me an explicit example to why the zero vector must be contained?
Best Answer
If a subspace is non-empty and satisfies the scalar multiplication axiom, then the zero vector comes "for free", since it is just any nonzero vector multiplied by the zero scalar. (In particular, any non-empty set that doesn't contain zero cannot possibly be closed under scalar multiplication. It's harder, but not all that hard, to show that lines/planes that don't go through the origin aren't closed under addition, either).
Furthermore, if you do not insist on the zero vector axiom, you will find that all the other axioms are conditions on the vectors in the space – e.g. that for every pair of vectors, they have a sum in the space – and hence are trivially satisfied by a set with no vectors. So the zero vector axiom is actually the only axiom that forces a vector (sub)space to be non-empty.
Hence, a vector (sub)space containing a zero vector is equivalent (in the presence of the other axioms) to it containing any vector at all. Since the zero vector is usually the easiest to find anyway, it's convenient to use its existence as an axiom.