Okay sorry. If I have a vector
\[\vec{x} = \left(\begin{matrix} x_1 \\ x_2 \end{matrix}\right) \]
that I want to express as a vector y with respect to the basis vectors v1 and v2, then
\[ \vec{x} = [\vec{v_1} \space \vec{v_2}]\vec{y} = V\vec{y} \]
so,
\[\vec{y} = V^{-1}\vec{x} \]
So, the linear transformation
\[T(x) = A\vec{x} \]
can also be written as
\[T(y) = AV\vec{y} \]
But at the end of the day, this quantity
\[ \left( AV\vec{y}\right) \]
is a vector with respect to the basis V. To make it a vector with respect to x again, I must apply V inverse:
\[T(x) = V^{-1}AV\vec{y} = B\vec{y} \]
so in conclusion, long story short,
\[B = V^{-1}AV \]