I recently started learning about dual basis vectors.
In a textbook, the author says that writing 2D vectors in the form of $$\vec A = A^x \vec e_1 +A^y \vec e_2=A_x \vec e^1 + A_y \vec e^2$$
serves to emphasize an important fact:
You can combine contravariant components with covariant basis vectors, or combine covariant components with contravariant basis vectors to define a quantity (such as vector $\vec A$) that is invariant under transformation of coordinates (e.g. rotation of coodinate system).
He then proceeded to say that the above statment should seem reasonable because covariant quantitites transform using a direct transformation matrix $D$ and contravariant quantities transform using an inverse transformatix $I$, with the effect $$D=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} I=\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$$$$DI=1$$ which is the identity matrix. So combining such quantities guarantees that the result is unaffected by transformation of coordinates.
I am finding it hard to see how that is reasonable. Is there an explicit example that shows what the author actually means?