I have some trouble with notation of bilinear functions. I will state the theorem i have trouble with:
A function $$\;\Bbb R^m\times \Bbb R^n\to\Bbb R\;$$ is bilinear if and only if it can be written in the form y=x1'Ax2 with A in Mmxn
Now, i will explain what i think is going on here. So I think what is meant with $$\;\Bbb R^m\times \Bbb R^n\to\Bbb R\;$$ is that a function whose domain is two arbitrarily vectors of dimension m and n respectively is mapped to a real number. For this to hold it should be possible to write it in y=x1'Ax2 where y is a real number x1' is a transposed vector and x2 is also a vector.
Now is this interpretation correct, that is, does $$\;\Bbb R^m\times \Bbb R^n\to\Bbb R\; $$represent any two arbitrary vectors as input, and is the output of this function a real number? Also, $$\;\Bbb R^m\times \Bbb R^n\;$$ specify nothing with respect to the vector being a column or row vector right?
If any of my interpretation is wrong please tell me, and possibly make things clearer for me.
Thanks in advance