Difference between coordinate space and vector space I am new to linear algebra and was going through this link, and it made me wonder:

Is there a technical difference between coordinate space (e.g., $R^n$) and vector space? If not, then why are there 2 terms for the same concept?

 A: Recall that if a vector space has a finite basis it is said to be finite dimensional, and the dimension is defined to be the number of vectors that make up this basis. Basis are (possibly finite) sets of vectors that span the vector space and are linearly independent. One can prove that every vector in said vector space can be written in one and only one way as a linear combination of these basis vectors. Say $V$ is a $K$ vector space with basis $B=\{v_1,\ldots,v_n\}$. Then if we have $$v=\alpha_1v_1+\cdots+\alpha_nv_n$$
we write $(v)_B=(\alpha_1,\ldots,\alpha_n)$ and say $v$ has coordinates $(\alpha_1,\ldots,\alpha_n)$ in the basis $B$. This immediately gives a mapping $V\to F^n$ given by $$v\mapsto (v)_B$$
This is the same as mapping each basis vector $v_i$ to $$(0,0,\ldots,\underbrace{1}_i,\ldots,0)$$
which entirely determines the transformation.
Note that $0\mapsto (0,0,\ldots,0)$; that $(v+w)_B=(v)_B+(w)_B$ and $(\lambda v)_B=\lambda (v)_B$ so this is a linear transformation, which gives an isomorphism between $V$ and $F^n$. This means $V$ and $F^n$ are essentially the same as vector spaces, that is, "there is only one vector space of dimension $n$ over a field $F$ up to isomorphism."
A: For a given dimension $n$ and a given field $k$, there is only one vector space of dimension $n$ over $k$ up to isomorphism, and this vector space is $k^n$, called the coordinate vector space.
