matrix multiplication by column vectors If $A$ is an $n \times n$ matrix, and $x$ and $y$ are column vectors in $R^n,$ show that $(A x)\cdot y=x \cdot (A^t y),$ and then that $(A x) \cdot (A y)= x(A^t A y).$ I tried to take $2 \times 2$ or $3 \times 3$ matrices and it worked but i did not know how to generalize it. thank you

hint. here are things you need to know:
(a) $x \cdot y = x^Ty = y^Tx,$
(b) $(Ax)^T = x^TA^T$
(c) $A(BC) = (AB)C$
i will do the first one. $Ax \cdot y = (Ax)^Ty = x^TA^Ty = x.A^Ty$
can you see how to do the second one?
