On transformation partly preserving a nonsingular bilinear function

Let F be an arbitrary field, and let Vn(F) be an n-dimensional vector space over F. We say that ϕ, a transformation on Vn(F), partly preserves a nonsingular bilinear function f(·, ·) : Vn(F) × Vn(F) → F if there exists a basis {ε1,ε2, … ,εn} for Vn(F) and a map π : {ε1,ε2, … ,εn} → Vn(F) such that f(α, εj) = f(ϕ(α), π(εj)) ∀α ∈ Vn(F), j = 1,2, … ,n. Let Mm×n(F) be the vector space of m × n matrices, and let Mn(F) be the vector space of n × n matrices over F. We prove that a transformation ϕ on Vn(F) is an invertible linear transformation if and only if ϕ partly preserves a nonsingular bilinear function. Then we characterize transformations ϕ on Mn(F) which satisfy det(ϕ(A)) = det(A) and Tr(ϕ(A)ϕ(B)) = Tr(AB). Finally we characterize the transformation groups Wm×n(F) = {ϕ∣ϕ(A) = PAQ ∀A ∈ Mm×n(F), where P ∈ GLm(F), Q ∈ GLn(F), PPt = Em, QQt = En} on vector space Mm×n(F) by as few invariants as possible.