I have some trouble with notation of bilinear functions I wi

I have some trouble with notation of bilinear functions. I will state the theorem i have trouble with:

A function R^m x Rⁿ   ---> R is bilinear if and only if it can be written in the form y=x1\'Ax2 with A in M^mxn

Now, i will explain what i think is going on here. So I think what is meant with R^m x Rⁿ ---> 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 R^m x Rⁿ represent any two arbitrary vectors as input, and is the output of this function a real number? Also, R^m and Rⁿ 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

Solution

Your interpretation is correct. The function takes two vectors as input, one of them(x1) is in R_m and the other(i.e x2) is in R_n and gives ouput a real number. x1 and x2 are column vectors. So x1 has size mx1 and x2 has size nx1. Also x1\' has size 1xm. A has size mxn. So as a result, x1\'Ax2 will give a real number