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

Here's a bruteforce way:
Let $A\in\Bbb R^{n\times n}$ and $x,y \in \Bbb R^n$, then we have
$$(Ax)\cdot y = \sum_{i=1}^n (Ax)_iy_i= \sum_{i=1}^n \left( \sum_{j=1}^n A_{i,j}x_j\right)y_i = \sum_{i=1}^n \sum_{j=1}^n A_{i,j}x_jy_i= \sum_{j=1}^n \sum_{i=1}^n A_{i,j}y_ix_j = \sum_{j=1}^n \sum_{i=1}^n A^T_{j,i}y_ix_j = \sum_{j=1}^n \left(\sum_{i=1}^n A^T_{j,i}y_i\right)x_j = \sum_{j=1}^n (A^Ty)_jx_j = (A^Ty)\cdot x $$
Applying this for $(Ax)\cdot z$ with $z = Ay$ we get 
$$(Ax)\cdot z = x \cdot A^Tz = x\cdot A^T(Ay)=x\cdot (A^TAy).$$
