Is $\mathbb{R}^n$ a vector space or a metric space?

Question

In my various courses, for instance, linear algebra and vector calculus, I am somewhat confused with what precisely $\mathbb{R}^n$ is.

From the definition of the Cartesian product, I would conceptualise $\mathbb{R}^n$ as the metric space with some distance operator, where all the points are just $n$-tuples. This is surely a distinct notion from vectors as isn't the point $A = (1,2,3)$, for instance, different from the vector $\vec{a} =\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}$ ? But if we were to consider the points in $\mathbb{R}^n$ as vectors then clearly it is a vector space. However I don't know whether these two conceptions of $\mathbb{R}^n$ are actually equivalent. Surely the vectors do not correspond to a specific point in space, unlike the points in $\mathbb{R}^n$.

Forgive me if this is a silly question, or if my question seems garbled. Also please help me with tags if they are inappropriate.

Answer 1

The definition of a vector space is "a set together with an addition and a scalar multiplication for which the following properties hold..."

By itself (as a set) $\mathbb{R}^n$ is not a vector space or a metric space. When you lump it in with appropriate operations, then the triple $(\mathbb{R}^n,\cdot,+)$ is a vector space. When you lump it in with an appropriate distance function, then $(\mathbb{R}^n, d(\cdot,\cdot))$ then you have a metric space.

$\mathbb{R}^n$ is just a set. You have to add this or that structure to make it anything else.

Answer 2

It is both a vector space and a metric space. Yes, a space can be both of them! And even more interestingly, it's an inner product space which means that we can measure angles in it. This is why $\mathbb{R}^n$ is so useful in geometry.

It's a vector space from an algebraic point of view. It's a metric space from an analytic point of view. Both of these views work together, hand in hand, to make mathematics more interesting than just a bunch of abstract definitions.