Artin's definition of "coordinate vector" This question sounds obvious, but I'm having difficulty understanding Artin's definition of "coordinate vector." Here is the relevant paragraph of Chapter 3, with the definition bolded.

Though row vectors take up less space, the definition of matrix multiplication makes column vectors more convenient, so we usually work with them. To save space, we sometimes use the matrix transpose to write a column vector in the form $(v_1, \ldots, v_n)^t$. As mentioned in Chapter 1, we don't distinguish a column vector from the point of $\mathbb{R}^n$ with the same coordinates. Column vectors will often be denoted by lowercase letters such as $v$ and $w$, and if $v$ is equal to $(a_1, \ldots, a_n)^t$, we call $(a_1, \ldots, a_n)^t$ the coordinate vector of $v$.

Right above this, Artin says that he doesn't distinguish between a point of $\mathbb{R}^n$ and a column vector, so I should have
$$
(a_1, \ldots, a_n) = (a_1, \ldots, a_n)^t,
$$
even though these are written differently. If $v = (a_1, \ldots, a_n)^t$, it sounds weird to say this is the "coordinate vector of $v$," since it is, in fact, $v$ itself.
Can someone help me understand the distinction?
 A: The coordinate vector of $v\in {\bf R}^n$ in the canonical basis is indeed $v$ itself. Recall that the canonical basis is the basis of ${\bf R}^n$ given by
$$
\pmatrix{1 \cr 0 \cr 0 \cr \vdots \cr 0\cr},  \qquad \pmatrix{0 \cr 1 \cr 0 \cr \vdots \cr 0 \cr}  \cdots \pmatrix{0 \cr 0 \cr 0 \cr \vdots \cr 1\cr}.
$$
After a while, you will work in other basis, for example eigenvector basis when studying diagonalisation of matrices. In that case, the coordinate vector of $v$ in these new basis will be different. Coordinates are relative to a given basis.
A: The problem is that Artin does not give a proper definition of $\mathbb R^n$. It does not even occur in the "Notation" section at the end of the book.
He only defines matrices, row vectors as $1 \times n$ matrices and column vectors as $m \times 1$ matrices. Here are two quotations which come closest to a definition of $\mathbb R^n$:

*

*In most of this book, we won’t make a distinction between an $n$-dimensional column vector and the point of $n$-dimensional space with the same coordinates. In the few places where the distinction is useful, we will state this clearly. [page 2]


*The column vector $e_i$ , which has a single nonzero entry $1$ in the position $i$, is analogous to a matrix unit, and the set $\{e_1, \ldots, e_n\}$ of these vectors forms what is called the standard basis of the $n$-dimensional space $\mathbb R^n$ (see Chapter 3, (3.4.15)). [page 9]
I would understand that the $n$-dimensional space mentioned in 1. is  $\mathbb R^n$, and in fact 2. supports this. But it does not become really clear what the distinction between elements of $\mathbb R^n$ and column vectors with $n$ coordinates should be.
Usually one writes elements $\mathbf a \in \mathbb R^n$ as $n$-tuples $(a_1,\ldots, a_n)$, but I think we do not need to regard such a tuple  as a matrix. Technically we could understand $\mathbf a$ as function $\mathbf a : \{1,\ldots,n\} \to \mathbb R$; then $a_i = \mathbf a(i)$. The real numbers $a_i$ can be arranged as tuples, row vectors or column vectors (or in some other form). If we agree to use column vector representation as a standard, then $(a_1 \ldots a_n)^t$ (note that I ommitted the separating  commas because matrices are written without commas) is the column vector representation of the $n$-tuple $(a_1,\ldots, a_n)$.
The sentence

Column vectors will often be denoted by lowercase letters such as $v$ or $w$, and if $v$ is equal to $(a_1 \ldots a_n)^t$, we call  $(a_1 \ldots a_n)^t$ the coordinate vector of $v$.

is indeed confusing. I would understand it as follows: $v$ denotes an abstract column vector, and if we write down its coordinates $a_i$, then we get the concrete representation $v = (a_1 \ldots a_n)^t$. This is called the the coordinate vector of $v$.
Perhaps it is also useful to have a look at Is there a reason vectors in space are represented as column vectors (in that nothing works with row vectors)?.
