Meaning of, and how to verify, a vector space *over* $\mathbb{R}$ In Axler's book on Linear Algebra he writes ($\mathbb{F}$ is here either $\mathbb{R}$ or $\mathbb{C}$):

The scalar multiplication in a vector space depends upon $\mathbb{F}$. Thus when we need to be precise, we will say that $V$ is a vector space over $\mathbb{F}$ instead of saying simply that $V$ is a vector space. For example, $\mathbb{R}^n$ is a vector space over $\mathbb{R}$, and $\mathbb{C}^n$ is a vector space over $\mathbb{C}$.

What is actually meant by $\mathbb{R}^n$ being a vector space over $\mathbb{R}$ and how can one verify that it is? What are all the implications of this? Is it just that any scalar multiplication in $\mathbb{R}^n$ depends upon numbers in $\mathbb{R}$?
To me, $\mathbb{R}^n$ feels larger than $\mathbb{R}$, so I find the wording, that one is over the other, hard to digest...
 A: In fact, $\mathbb{C}^n$ is an $n$-dimensional vector space over $\mathbb{C}$, but $\mathbb{C}^n$ is also a $2n$-dimensional vector space over $\mathbb{R}$.  When you understand why that is so, then you will know the answer to your question.
You can find a set $\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}$ of $n$ vectors such that every vector $v\in\mathbb{C}^n$ can be expressed as a linear combination $c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n$, where $c_1,\ldots,c_n$ are complex numbers.
You can also find a set $\{\mathbf{v}_1,\ldots,\mathbf{v}_{2n}\}$ of $2n$ vectors such that every vector $v\in\mathbb{C}^n$ can be expressed as a linear combination $c_1\mathbf{v}_1+\cdots+c_{2n}\mathbf{v}_{2n}$, where $c_1,\ldots,c_{2n}$ are real numbers.
Thus any point in $\mathbb{C}^n$ can be specified by $n$ complex numbers or by $2n$ real numbers.
The set $n$ vectors $(1,0,\ldots,0,0)$, $(0,1,0,\dots,0)$,$(0,0,1,0,\ldots,0),\ldots,(0,0,\ldots,0,1)$ can serve in the first role above, where the scalars by which they will be multiplied are complex numbers.
The $2n$ vectors $(1,0,\ldots,0,0)$, $(0,1,0,\dots,0)$,$(0,0,1,0,\ldots,0),\ldots,(0,0,\ldots,0,1)$, $(i,0,\ldots,0,0)$, $(0,i,0,\dots,0)$,$(0,0,i,0,\ldots,0),\ldots,(0,0,\ldots,0,i)$ can serve in the second role above, where the scalars by which they will be multiplied are real numbers.
A: When you give the axioms of a vector space $V$, you have to talk about a certain field: for example, in the axioms that talk about scalar multiplication, you'll say something like "for every $x \in \mathbb{F}$ and every $v \in V$ there's an element $xv$ of $V$....".  In this case we say that "$V$ is a vector space over $\mathbb{F}$".
So you're right: $V$ is a vector space over $\mathbb{F}$ means that the scalars you can multiply by are elements of $\mathbb{F}$.  It's just a conventional use of the word over.
The reason to mention it is that sometimes which field you are working over is not obvious.  Take $\mathbb{C}^n$.  It is a vector space over $\mathbb{C}$ in the obvious way, but it is also a vector space over $\mathbb{R}$ (if you can use elements of $\mathbb{C}$ as scalars then you can certainly use $\mathbb{R}$ as $\mathbb{R} \subset \mathbb{C}$).  What's more it has dimension $n$ over $\mathbb{C}$ but $2n$ over $\mathbb{R}$ so it really matters which field you work over.
