Consider a vector $V$ in two coordinate systems related by a rotation for example. Let $R$ be the rotation matrix in $2D$. We are considering a $2D$ coordinate system and cartesian coordinates here. In the two different systems we can write :
$$V=v^xe_x+v^ye_y=\tilde{v}^{\tilde{x}}\tilde{e}_{\tilde{x}}+\tilde{v}^{\tilde{y}}\tilde{e}_{\tilde{y}}$$
This is because the components and the basis of the vectors would transform in an opposite manner, such that the net object remains invariant. From the transformation rules of co-variant and contravariant objects, we can see that if we collect the components and the basis in a tuple, then we can write :
$$\begin{bmatrix} \tilde{v}^{\tilde{x}} \cr \tilde{v}^{\tilde{y}} \end{bmatrix} = R \begin{bmatrix} v^x \cr v^y \end{bmatrix}$$
If we ignore the basis for a second, these are nothing but the new components, after rotation.
Similarly,
$$\begin{bmatrix} \tilde{e}_{\tilde{x}} \cr \tilde{e}_{\tilde{y}} \end{bmatrix} = R \begin{bmatrix} e_x \cr e_y \end{bmatrix}$$
This makes sense, as we can now show :
$$\tilde{v}^{\tilde{x}}\tilde{e}_{\tilde{x}}+\tilde{v}^{\tilde{y}}\tilde{e}_{\tilde{y}}=[\tilde{v}^{\tilde{x}}\space\space \tilde{v}^{\tilde{y}}]\begin{bmatrix} \tilde{e}_{\tilde{x}} \cr \tilde{e}_{\tilde{y}} \end{bmatrix} = [ v^x\space\space v^y] R^{-1} R \begin{bmatrix} e_x \cr e_y \end{bmatrix} = [ v^x\space\space v^y] \begin{bmatrix} e_x \cr e_y \end{bmatrix}=v^xe_x+v^ye_y$$
In this sense, we can see that components and basis transform in an opposite manner. This seems pretty general, where we have not assumed anything about what these vectors look like, and all that.
In most physics books, this is shown in a different manner, and I'm unable to reconcile the two, even though they both give the same answer. In most physics texts, the tuple of components of the vectors are treated as 'column' vectors, and they transform in the same way as I've shown above. The basis vectors are represented by column matrices, and the components are just numbers. Combining both we get a column matrix, which now represents the vectors. This is also termed as an active transformation, where the transformation acts on the vector, or more appropriately, the column matrix whose elements are the components of the vector.
The difference comes when we describe how the basis transforms. In most books, the basis is also described as a column vector, and the transformation is given as :
$$\tilde{e}_{\tilde{x}} = R^{-1} e_x$$
$$\tilde{e}_{\tilde{y}} = R^{-1} e_y$$
$R$ is the same rotation matrix, but the basis are now column vectors too. This kind of transformation is now called passive transformation, and if I plug in the numbers, it turns out that the two rules of transforming the basis vectors give the same answer.
In the first case, I've taken the basis vectors as components of tuple, and acting $R$ on them, gives the new tuple, containing the new basis vectors. In the second case, I take the basis vectors as column vectors, and directly applying $R^{-1}$ to it, gives me the new basis vectors.
This second 'physics' method of doing this, makes me a bit uncomfortable. Can someone prove that the two methods are equivalent, without explicit calculations. The two ways of defining the transformation of the basis vectors, seem to give the same answer. How is this physics way of transforming a basis vector related to the mathematics way.