As I understand it, if I just write $\mathbf{v} = \begin{bmatrix} a & b & c \end{bmatrix}^T$ it is assumed that the components, $a$, $b$, and $c$ refer to the coefficients of the standard basis vectors in the linear combination of them that produces $\mathbf{v}$.
Whereas if I write $\left[\mathbf{v}\right]_B = \begin{bmatrix} a & b & c \end{bmatrix}^T$ then it's understood that the components of $\mathbf{v}$ refer to the coefficients of the vectors in the basis $B$ in the linear combination that produces $\mathbf{v}$.
But in general, if I'm told that there is a vector space $V$ that has a basis, $C$, do I still assume that $\mathbf{v} = \begin{bmatrix} a & b & c \end{bmatrix}^T$ refers to the standard basis, rather than $C$? I suppose this also hinges on whether the standard basis always exists? (Seems like it must for finite dimensional real vector spaces, just keep having vectors with only a single 1 and all other components 0, but I don't know if that generalizes to the whole menagerie of infinite dimensional spaces and arbitrary fields)