Verify basis for R² and find coordinate vector [v]_β

[karush]

Verify that
$\beta=\left\{\begin{bmatrix}
0\\2
\end{bmatrix}
,\begin{bmatrix}
3\\1
\end{bmatrix}\right\}$
is a basis for $\Bbb{R}^2$
Then for $v=\left[
\begin{array}{c}6\\8\end{array}
\right]$, find $[v]_\beta$
ok, I presume next is
$c_1\begin{bmatrix}
0\\2
\end{bmatrix}
+c_2\begin{bmatrix}
3\\1
\end{bmatrix}= \left[
\begin{array}{c}6\\8\end{array}
\right]$
by augmented matrix we get (the book did this?)
$\left[ \begin{array}{cc|c} 0 & 3 & 6 \\ 2 & 1 & 8 \end{array} \right]
=\left[ \begin{array}{cc|c} 1 & 0 & 3 \\ 0 & 1 & 2 \end{array} \right]$
hence
$[v]_{\beta}=\left[
\begin{array}{c}3\\2\end{array}
\right]$
following an example I don't think I understand the notation of $[v]_{\beta}$

 

[Evgeny.Makarov]

$[v]_\beta$ denotes the coordinates of vector $v$ in basis $\beta$.

Could you say from which textbook this notation is taken?

 

[karush]

https://www.physicsforums.com/attachments/9053
this one