So I aced linear algebra over the fall semester, though I'm deeply troubled in that I struggle to really describe what I did. I cannot say with confidence what it all meant, nor do I have any sort of intuition with the subject. I don't think i ever really "got it" or internalized it. Therefore, while i still have a week before summer classes, I started re-reading the text to gain some insights.
I'm starting with subspaces and am struggling with why a subspace must contain the zero vector. The closure under linear combinations is something i understand, being that my go to example is a plane in $\mathbb{R}^3$. But why must $ \mathbf{0} \in V $?
A subspace must be closed under scalar products. And, a subspace must be a non-empty subset. So, if you have a subspace, then you have at least one vector $v$ in it. Then, you also have the scalar product $0\cdot v$ in the subspace. But, it follows from the distributivity axioms in a vector space, $0\cdot v=0$ always. Thus, $0$ must be in the subspace.