Bilinear Form - Coordinate Representation

Coordinate Representation

Let V ≅ Fn be an n-dimensional vector space with basis {e₁, ..., e_n}. Define the n × n matrix A by A_ij = B(e_i, e_j). If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:

Suppose {f₁, ..., f_n} 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.