Coordinate Representation
Let V ≅ Fn be an n-dimensional vector space with basis {e1, ..., en}. Define the n × n matrix A by Aij = B(ei, ej). If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:
Suppose {f1, ..., fn} is another basis for V, such that:
- = S
where S ∈ GL(n, F). Now the new matrix representation for the symmetric bilinear form is given by: STAS.
Read more about this topic: Bilinear Form