Proof that Determinant is Scale Factor

Question

I've seen a lot of supposed properties of linear transformations that're never proven -- just often repeated. These include:

1. The determinant is the scale factor between the volume of region in your space and the volume of the image of that region. This apparently applies no matter the shape of the region.
2. A linear transformation always takes parallelograms/ higher analogs to parallelograms/ higher analogs.
3. A linear transformation always takes ellipses to ellipses.

How would I go about proving these things?

Answer 1

Note that each Characteristic value (eigenvalue) is the scale factor for the unit vector in the original orthogonal coordinate system. Next, Det(M) is the same whether it is diagonalized or not. so Det(M)=Product of e/vals= product of scale factors for each original dimension, hence the product is the ratio of the new volume to the old.