The arena vectors live in
A vector space is just the full playground of vectors you're working with, together with the two moves you already know. The one rule that makes it a *space*: it must be closed under adding and scaling. Combine any vectors inside it, and the result is still inside it — you can never accidentally fall out.
Subspaces: rooms within the space
A subspace is a smaller vector space sitting inside a bigger one — like a flat sheet inside a room. The catch: it must pass through the origin, because scaling any vector by 0 lands you at (0, 0), and a closed space has to contain that point. A line or plane through the origin is a subspace; one that misses the origin is not.
line through origin: all multiples of (2, 1) -> subspace line NOT through origin: points (2t, 1t + 5) -> NOT a subspace
You've already met subspaces without the name: the span of any vectors through the origin is always a subspace. So 'span' and 'subspace' are two views of the same flat, well-behaved shape.
Vectors that aren't arrows
Here's the surprise that makes the subject powerful: anything you can add and scale sensibly is a vector. Polynomials qualify — add 2x + 1 and 3x + 4 to get 5x + 5, scale by 2 to get 4x + 2. Sound waves and functions qualify too. The arrow was only ever a helpful picture.
A gentle nod to dimension
How big is a space? The dimension counts the number of independent directions you genuinely need to reach everything in it. A line is 1-dimensional, a flat plane is 2-dimensional, the room around you is 3-dimensional. We'll make this precise later — for now, just hear it as 'how many real choices you have.'
That word *independent* points straight at the redundancy you met last guide. Strip away every redundant direction, count what's left, and you have the dimension. It's the perfect bridge into the next rung of this ladder.