basis of vector space : what mean $v=(1,2,3)$ in $\mathbb R^3$? I have a small doubt. Consider the vector space $V=\mathbb R^3$ and $\{v_1,v_2,v_3\}$ a basis of $V$. 
Q1) What exactly mean $$v=\begin{pmatrix}1\\2\\3\end{pmatrix}\in \mathbb R^3.$$
Do we interpret it as $$v=1v_1+2v_2+3v_3,$$
or is it really $$v=1e_1+2e_2+3e_3$$ where $\{e_1,e_2,e_3\}$ is the canonical basis ? 
Q2) By the way, my question is may be stupid, but if we are in $\{v_1,v_2,v_3\}$, don't we have that $$v_1=\begin{pmatrix}1\\0\\0\end{pmatrix},\quad v_2=\begin{pmatrix}0\\1\\0\end{pmatrix}\quad \text{and}\quad v_1=\begin{pmatrix}0\\0\\1\end{pmatrix},$$
and thus $\{v_1,v_2,v_3\}$ is the canonical basis with respect to it self ? 
 A: $(1,2,3)=v_1+2v_2+3v_3$. 
No, $\mathfrak B=\{v_1,v_2,v_3\}$ may not be the canonical basis. Yes, the coordinates of $v_1$ with respect to $\frak B$ are $(1,0,0)$ but for $\frak B$ to be the canonical basis, the coordinates of $v_1$ with respect to the canonical (standard) basis $\{e_1,e_2,e_3\}$ must be $(1,0,0)$. Similarly for $v_2,v_3$.
For example, the set $S=\{(1,0,0),(0,1,0),(1,0,1)\}$ is a basis of $\Bbb R^3$ since $v_i$ are linearly independent (check) and span $\Bbb R^3$. In this coordinate system, $(1,2,3)=1(1,0,0)+2(0,1,0)+3(1,0,1)$ which is $(4,2,3)$ in the canonical basis. 
Clearly, the coordinates of $v_3$ are $(0,0,1)$ with respect to $S$ but with respect to the standard basis, its coordinates are $(1,0,1)$. Thus this basis is not canonical.
A: I would say that $v = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ means $v = e_1 + 2e_2 + 3e_3$, where $e_1, e_2, e_3$ are the standard basis vectors. If I wanted to express $v = v_1 + 2v_2 + 3v_3$, I would either write exactly that, or I would write
$$[v]_B = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},$$
where $B = (v_1, v_2, v_3)$.
As a personal preference, I tend to reserve column vectors to be coordinate vectors. If my underlying vector space was $\Bbb{R}^3$ anyway, I would refer to its elements as ordered $n$-tuples, such as "$(1, 2, 3)$". If I wanted to convert $v = (1, 2, 3)$ into a column vector (such as for multiplying to a matrix), I prefer to write
$$[v]_S = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},$$
where $S$ is the standard basis.
I prefer this convention because I think it helps separate out this confusion. Too many times have I come across students who find a matrix for a linear transformation between spaces of polynomials, and ask how to multiply a polynomial to a matrix! If column vectors only ever appear as coordinate vectors, it means that even when the underlying space is $\Bbb{R}^n$, they still always have to compute the coordinate vector (rather than just put the numbers in a column - an approach that fails when you move away from standard bases).
All that said, it is possible that the given $v$ really does represent $v_1 + 2v_2 + 3v_3$. That would be, in my mind, a hefty abuse of notation, but I've seen some truly shocking instruction in linear algebra in my time...
A: 1) From the context, $v = v_1 +2 v_2 + 3 v_3$. It would make no sense to have $v$ expressed in the canonical basis as another basis is introduced.
2) Your question is not stupid, and you are totally right.
