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.

A subspace has to be a vector space and therefore must be closed under scalar multiplication. There is always a $0$ value in the field of scalars, so $0\cdot\vec v$, which is the origin $\vec0$, has to be in the subspace.
You might be familiar with thinking of Euclidean vector spaces like $\mathbb R^3$, where the nontrivial subspaces are lines or planes through the origin. The collection of lines or planes that don’t necessarily go through the origin are called “affine spaces” or “affine subspaces,”* but those are not vector spaces if they don’t go through the origin.
*Like sometimes happens with mathematical definitions more than in English, the adjective “affine” is not restrictive; affine subspaces are not necessarily subspaces.
