# Are tensors always invariant?

I'm new to tensors and what I've learned about them is that they are objects that are 'invariant' under transformations, is this true for all ranks of tensors? If I got $$n$$ functions $$A^i$$ where $$i = 1,2,3....,n$$ in the $$S$$ coordinate for example and would like to transform these functions to $$\bar A^i$$ to the $$\bar S$$ coordinate, by the law of transformation:

$$\bar A^i = \frac{\partial{\bar x^i}}{\partial{x^j}} A^j$$ [1st rank tensor transformation]

But according to my understanding, if tensors are invariant the tensor $$A^i$$ before the transformation should equal $$\bar A^i$$ , is this correct? Do I even make sense in anything? Thanks