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[Community Question] Linear-algebra: Is $\mathbb{R}^n$ a vector space or a metric space?

One of our user asked:

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.


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