Let V be a vector space of dimension n over a field K. A map $${\displaystyle B:V\times V\rightarrow K}$$ is a symmetric bilinear form on the space if:
• $${\displaystyle B(u,v)=B(v,u)\ \quad \forall u,v\in V}$$ See more

Let $${\displaystyle C=\{e_{1},\ldots ,e_{n}\}}$$ be a basis for V. Define the n × n matrix A by $${\displaystyle A_{ij}=B(e_{i},e_{j})}$$. … See more

A basis $${\displaystyle C=\{e_{1},\ldots ,e_{n}\}}$$ is orthogonal with respect to B if and only if :
$${\displaystyle B(e_{i},e_{j})=0\ \forall i\neq j.}$$ See more

Let V = R , the n dimensional real vector space. Then the standard dot product is a symmetric bilinear form, B(x, y) = x ⋅ y. The matrix corresponding to this bilinear form (see below) on a standard basis is the identity matrix.
Let V be any vector … See more

Two vectors v and w are defined to be orthogonal with respect to the bilinear form B if B(v, w) = 0, which, for a symmetric bilinear form, is equivalent to B(w, v) = 0.
The radical of a bilinear form B is the set of vectors orthogonal with every vector in V. That this is a … See more

Let B be a symmetric bilinear form with a trivial radical on the space V over the field K with characteristic not 2. One can now define a map from … See more