Q:

Exact number of vectors in a vector space

I was doing some practice questions in my book and I was asked T/F if there are real vector spaces containing exactly seven vectors. I was thinking that this is True, but the books tells me that the right answer is False. Why is this so? Thanks!

A:

Every vector space (besides the trivial zero space) has infinitely many vectors in it, since for any vector in the vector space,  we can always find another vector in the vector space by multiplying by a real number (this is because vector spaces are closed under scalar multiplication), and we know that there are infinitely many real numbers, so it follows that there would be infinitely many vectors.
If I told you that $\vec{u}\in V$, and I asked you to tell me another vector in the space, you could tell me $2\vec{u}$ or $3\vec{u}$ or $2.333344\vec{u}$, etc...

