Math Help - Permutations Matrices Question

This is a homework problem i had. Help!

Explain why the dot product of x and y equals the dot product of Px and Py.
Then From (Px)^T(Py)=x^Ty deduce that P^TP=I for any permutation. With x=(1,2,3) and y=(1,4,20) choose P to show that Pxdoty is not always xdotPy.

I dare say that you have most of us at a disadvantage.
I for one have absolutely no idea what that question means.
You must define the terms and explain the notation.

I think its referring to the Identity Matrix I. and the different permutations which are P. And ^T means to find the dot product of

P is a "permutation" matrix: every row and column contains exactly one "1" and all other entries are 0. For example,

just permutes the entries x, y, and z.

If, instead of we use then .


Essentially since both x and v are permuted in the same way, the "corresponding terms" still match up.

However,

while

so is not, in general, equal to .