# What isn’t a vector space?

I’m really confused about vector spaces. We’re learning about them in Linear Algebra, and my book doesn’t give good examples of what a vector space is. I understand sets and vectors, but I don’t understand vector spaces. From the definitions they’ve provided so far, it seems anything can be a vector space.

Can someone provide a simple example of what isn’t a vector space so I can make a distinction?

First off, a vector space needs to be over a field (in practice it’s often the real numbers $\Bbb R$ or the complex numbers $\Bbb C$, although the rational numbers $\Bbb Q$ are also allowed, as are many others), by definition. Thus, for instance, the set of pairs of integers with the standard componentwise addition is not a vector space, even though it has an addition and a scalar multiplication (by integers) that fulfills all of the properties we ask of a vector space.
A vector space needs to contain $\vec 0$. Thus any subset of a vector space that doesn’t, like $\Bbb R^2 \setminus \{\vec 0\}\subseteq \Bbb R^2$ with the standard vector operations is not a vector space. Similarily, a vector space needs to allow any scalar multiplication, including negative scalings, so the first quadrant of the plane (even including the coordinate axes and the origin) is not a vector space.
A more subtle example is the circle (with some chosen zero) where addition is done by adding distances along the circle from the chosen zero (equivalently by adding angles), and scalar multiplication is done by multiplying distances (angles). Here we get into trouble with scalar multiplication again, because the zero vector is simultaneously representing $360^\circ$, so what should $0.5$ multiplied by that vector be? $0^\circ$? $180^\circ$? It would be both at the same time, which is not good.