Let's say I have a set of vectors $$a, b, c, d$$ that are all n elements.
So if I multiply the dot product of two vectors (one of which is a transpose) I will receive a scalar.
$$(a\cdot b^T)*(c\cdot d^T)=e $$
Where e is a scalar. Also the dot product gives me scalar as well.
In the hw problem I am doing, I know that $$b^T \cdot c$$ would give me a dot product I want that I can move as a scalar to another section of the problem.
So, could $$(a\cdot b^T)*(c\cdot d^T)=(a\cdot d^T)*(c\cdot b^T)$$???
If so, why? Because they are in the same field?
I am tempted to prove it by saying the sum of the elements are all the same:
$$(a\cdot b^T)*(c\cdot d^T)=\sum_{i=0}^n a_i*b_i*c_i*d_i$$
Does that make any sense?
Best Answer
No, you cannot do that. Take $a=(1,0)$, $b=(0,1)$, $c=(0,1)$, and $d=(1,0)$. Then$$(a.b^T)\times(c.d^T)=0\text{, but }(a.d^T)\times(c.b^T)=1.$$