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Let A be a real m x n matrix and let x be in

**R**^n and y be in

**R**^m (n and m dimensional real vector spaces, respectively). Show that the dot product of Ax with y equals the dot product of x with A^Ty (A^T is the transpose of A).

The way I went about starting this problem is to use the definitions where the definition of the dot product on real numbers is: the sum with k from 1 to n of ak * bk and the definition of matrix multiplication for the entries of Ax would each be of the form: sum from k=1 to k=n of Aik * xk1.

Hopefully that was clear enough, but what I come up with when I plug the second definition into the first is one sum inside another and it seems like either I'm missing something that I can simplify or I went in the wrong direction! Does anyone have any suggestions for what I could try here? Any help is appreciated