Removing tranpose by multiplying

Question

I have a problem understanding this rule, that my professor wrote up: $$\vec{x}^TB\vec{y}=\vec{x}\cdot(B\vec{y}),$$ when $x\in\mathbb{R}^k$ and $y\in\mathbb{R}^m$. I can't find an example where this is correct, fx. $$\left( \begin{array}{cc} a & b \\ \end{array} \right).\left( \begin{array}{cc} c & d \\ e & f \\ \end{array} \right).\left( \begin{array}{c} g \\ h \\ \end{array} \right)=\left( \begin{array}{c} (a c+b e) g+(a d+b f) h \\ \end{array} \right),$$ however, $$\left( \begin{array}{c} a \\ b \\ \end{array} \right) \left(\left( \begin{array}{cc} c & d \\ e & f \\ \end{array} \right).\left( \begin{array}{c} g \\ h \\ \end{array} \right)\right)=\left( \begin{array}{c} a (c g+d h) \\ b (e g+f h) \\ \end{array} \right)$$ does not give the same. Any idea on why it doesn't give the same?

Answer 1

When we write $\vec{v} \cdot \vec{w}$, we mean the dot product $\vec{v}^\top \vec{w}$. So $\vec{x}^\top B \vec{y}$ is literally the same as $\vec{x} \cdot (B \vec{y})$.

You should not use "$\cdot$" for usual matrix multiplication.

So you should have written $$\left( \begin{array}{cc} a & b \\ \end{array} \right)\left( \begin{array}{cc} c & d \\ e & f \\ \end{array} \right)\left( \begin{array}{c} g \\ h \\ \end{array} \right)=\left( \begin{array}{c} (a c+b e) g+(a d+b f) h \\ \end{array} \right),$$ which is correct, and the second one should be $$\left( \begin{array}{c} a& b \end{array} \right) \left(\left( \begin{array}{cc} c & d \\ e & f \\ \end{array} \right)\left( \begin{array}{c} g \\ h \\ \end{array} \right)\right)= ( a (c g+d h) + b (e g+f h))$$