A subspace is just a smaller vector space sitting inside a bigger one. It’s a set of vectors closed under addition and scalar multiplication. Think of it as any line or plane through the origin that you can move around in freely, but you can’t “step out” of it by adding or scaling vectors.

A classic example of a subspace is all vectors in 3D space whose last component is zero, like (x, y, 0). This forms a flat plane through the origin. It contains the zero vector, always stays on the plane when you add vectors or multiply by numbers, and never escapes its own “clubhouse”.

An example of something that is not a subspace: all points in 2D space where x + y = 1. This is a line, but it doesn’t go through the origin, so it fails the subspace test—the zero vector (0, 0) isn’t on this line, and scalar multiplication can take you off the line.[5][3]

Q: What if the set is a line through the origin?
A: That's always a subspace—origin is included, and the other rules work out.

Q: Why does missing the origin break the rules?
A: The zero vector (all zeros) must always be part of every subspace. If not, it's out!

Q: Is the set of all positive vectors a subspace?
A: No—the sum of two positive vectors is positive, but multiplying by a negative number gives negatives, which aren’t in the set.

Q: So subspaces are “closed” under addition and multiplication?
A: Exactly! They keep everything in their “clubhouse.”