In an introductory course where we have seen that the span of n linearly independent vectors is a vector space, we are now supposed to show that the span of m < n linearly independent vectors is a vector space as well. As hints, we are supposed to use that the vectorspace axioms are still satisfied and that addition and scalar multiplication do not result in leaving the subspace.
My problem with this is that I do not really get what might be asked. Should I just write down the axioms from the lecture again and say "still applicable" beneath? Thanks for any clarification or idea on what might be asked by the instructors.
You're right that a lot of the work is redundant. Many of the conditions necessary to be a vector space inherit to subsets. For example, if $V$ is a vector space, then given any subset $W \subseteq V$, we have $u + v = v + u$ for all $u, v \in W$, since we have $u + v = v + u$ for all $u, v \in V$.
Indeed the only things you don't get for free is the existence of the additive identity, the existence of additive inverses, or the guarantee that the vector space operations (addition and scalar multiplication) still make sense when restricted to the subset. That is, you don't necessarily have closure, which would prevent the operations being operations on $W$.
As it turns out, you only really need to guarantee closure of the two operations on $W$, and exclude the possibility that $W = \emptyset$. If you guarantee these conditions, then $W$ must include the additive identity $0$, and must contain inverses (indeed, these follow from closure under multiplication by the scalars $0$ and $-1$).
As you will cover soon, such a subset is called a subspace of $V$. The term is either defined to be a subset $W$ which is a vector space when restricting addition and scalar multiplication to the subset, or it's defined in terms of simultaneously being non-empty and closed under addition and scalar multiplication. Either way, the definitions are equivalent.
You can expect your instructor to give you multiple questions about showing various subsets are subspaces. For example, they might ask you to prove or disprove whether, say $\{(x, y, z) \in \Bbb{R} : x^2 - y = 0\}$ is a subspace of $\Bbb{R}^3$.