Linear Algebra – Difference Between Abstract Vector Spaces and Vector Spaces

Question

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces.

I've searched a little and made a superficial comparison between both and found that they are the same thing. Is that correct or there's something I'm missing? Also, if it's correct, why two names to the same thing?

Answer 1

I am quite sure *vector spaces" and "abstract vector spaces" mean the same thing, and as Micah suggests, "abstract vector spaces" may simply make it more explicit that the spaces of concern are not necessarily $\mathbb C^n$ or $\mathbb R^n$. However, most courses and/or texts on linear algebra teach vector spaces as spaces which need not be $\mathbb R^n$ or $\mathbb C^n$.

For example, from Wikipedia, you can read:

Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are best thought of as abstract mathematical objects with particular properties ...

...Historically, the first ideas leading to vector spaces can be traced back as far as 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space...