As usual, it is useful to think about how a <bilinear form> looks like in terms of <vectors> and <matrices>.

Unlike a <linear form>, which <matrix representation of a linear form>[was a vector], because it has two inputs, the bilinear form is represented by a matrix $M$ which encodes the value for each possible pair of <basis vectors>.

In terms of that <matrix>, the form $B(x,y)$ is then given by:
$$
B(x,y) = x^T M y
$$


Matrix representation of a bilinear form

For the typical case of a linear form over <\R^n>, the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the <basis vectors>.


Ciro Santilli @cirosantilli 37

 Incoming links: Matrix representation of a linear form