While learning about Special Relativity I learnt that we use the Transformation matrix to alter the space .This matrix differs for $Contravariant$ and $Covariant$ vectors.Why does it happen?,Why one kind of matrix ($Jacobian$) for $basis$ $vectors$ and other kind($Inverse$ $Jacobian$) for $gradient$ ,$divergence$ etc.Why does the matrix change ,why does changing quantities require a new matrix and why does $Inverse$ $Jacobian$ satisfy this?
$My$ $understanding$ $of$ $how$ $Jacobian$ $achieves$ $this$:
Here Grant says that by moving a small dist $\partial x$ ,we produce a change in $\partial f_1$ direction and $\partial f_2$ direction , $\partial f_1$ in being $x+\sin(y)$ direction & $\partial f_2$ being in $y+\sin(x)$ ,(ie) $\hat{x}$ transforms to $x+\sin(y)$ and $\hat{y}$ axis transforms to $y+\sin(x)$, by moving a small distance $\partial x$ in $x$ axis in input we produce a effect on both the axes in output space, similar thing happens in y direction .
$$(x,y) \rightarrow (x+\sin(y),y+\sin(x)) $$
$$ J = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \\ \end{bmatrix} $$
Is my interpretation a right one?
And why does the $Jacobian $ Matrix change for covariant and contravariant vectors?