Suppose that
$$ \mathbf A =(A_1, A_2,A_3) $$ $$ \mathbf B =(B_2, B_2,B_3) $$ $$ \mathbf C =(C_3, C_2,C_3) $$
We know that $$ \mathbf{A \cdot B \times C} $$
is equal to
$$\begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z & \\ C_x & C_y & C_z\end{vmatrix}$$
However, suppose that we change the basis of the space to three arbitrary vectors, $$ \mathbf { \vec{a},\vec b,\vec c}$$
show that
$$ \mathbf{A \cdot B \times C} $$
is transformed into
$$\begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z & \\ C_x & C_y & C_z\end{vmatrix} (\mathbf{a \cdot b \times c)}$$
So is this transformation somewhat of a linear one, and how do I prove this statement? I tried with components, but I think there must be a most straightforward way that doesn't involve all of these componente.