Affine transformations

Can anybody tell me where I can find
the proof that any ellipse remains ellipse
after any affine transformation.

You can scale a ellipse to make a circle so …

Then again a circle is an ellipse with the 2 focal points being = .

Besides that, what can I say. A transformation will be applied to the equation, so why don’t you just proove it that way, for each case (translation, rotation, …)