/sci/ - Science & Math

If biliniear form is a matrix, it is a structure, an entity sandwiched between two vectors like in the picture and matrix multiplication is performed twice with each vector and the result is a scalar. Now imagine the dot product which is also a bilinear form but there is literally nothing in between, so the vectors touch each other directly. How should I think about this nothingness?? What is the "form" here, the dot product itself? But that is an operation, not an entity, not a "structure".