I was reading some notes on tensors and was confused by a particular section which I shall write in bold. The Paragraph is as follows:
In transforming between coordinate systems, a vector with contravariant components $A^j$ in the original (unprimed) coordinate system and contravariant components $A'^i$ in the new (primed) coordinate system transforms as $$A'^i = \frac{\partial x^{i'} }{\partial x^j}A^j$$
where the $ \frac{\partial x^{i'} }{\partial x^j}$ terms represent the components in the new coordinate system of the basis vectors tangent to the original axes.
Likewise, for a vector with covariant components $A_j$ in the original (unprimed) coordinate system and covariant components $A'_i$ in the new (primed) coordinate system, the transformation equation is $$A'_i = \frac{\partial x^j}{\partial x ^{i'}}A_j $$ where the $ \frac{\partial x^j}{\partial x ^{i'}}$ terms represent the components in the new coordinate system of the (dual) basis vectors perpendicular to the original axes.
If anyone can explain in more detail (perhaps by using a diagram if that would be more helpful) what exactly is meant by the sentences in bold, or if it can be illustrated by an example if you think that would be a good way of explaining.
Thanks a lot!