Linear Algebra intuition behind subspaces Hello I am trying to understand better the intuition behind subspaces. I am aware that the subspace of a vector space must have zero(go through the origin), has to be closed by addition, and has to be closed by multiplication. The aspect I fail to understand is why must it equal zero/go through the origin. If anyone has any analogy for subspaces I'd also love to hear it thank you.

Closed by multiplication $\implies$ zero must be in the subspace $U\subseteq V$, as stated at the answer above. Also, a subspace is, in particular, a vector space and vector spaces must have a neutral element: the vector $e\in U$ for which $u+e=u$ for all vectors $u\in U$. And by unicity of the neutral element of the "bigger" vector space $V$, the element $e\in U$ with that property must be the same element as the neutral of $V$, that is: zero.
Indeed, you only prove that zero is in $U$ ($U$ is the set you want to be a subspace) to conclude that $U\not = \emptyset$, because although $\emptyset$ is vacuosly closed by addition and multiplication, it doesn't have a neutral element (hence is not a vector space). So, on proving that $U$ is a subspace you can show that $0\in U$ or that $c \in U$ for some convenient constant vector (and show those "close properties")... Again, what really matters is that $U\not = \emptyset$.
