Transformations:

1.

2.

3.

which is indeed a bilinear mapping.

I don't think this is correct Sameed.

The three transformations to be carried out in order are:




Instead of doing each transformation on the result of the previous one, you have substituted for z each time. It's easy to see that it's not correct if you choose a particular case where A=2, B=0, C=1, D=0 and substitute into your final formula, which gives:

The first transformation in this case is
the second is
and the third which is the identity transformation.
So transformation 1 gives us and transformation 2 gives us
This is not what your formula gives.

if you do it in the following way you are less likely to get it wrong:






So and

Finally

which is a bi-linear transformation