dedekind::geometry::IsLinearMapModule Concept Reference

Verify that LinearMap<F, R, C> behaves as a module over scalar F. More...

Concept definition

template<>
concept dedekind::geometry::IsLinearMapModule = 

Detailed Description

Verify that LinearMap<F, R, C> behaves as a module over scalar F.

linear_map__Algebraic_Verification_for_Matrix_Scalars

When F satisfies IsMatrixScalar:

• the implementation provides the operations needed by this MVP
• those operations behave like a field only under idealized exact arithmetic
• in actual IEEE arithmetic, this is a field-like policy witness rather than a categorical field proof

Consequence: LinearMap<F, R, C> behaves as a vector-space-like carrier over F in the same operational sense used by HasFieldOperators.

This means LinearMap satisfies the four axioms of linear action:

1. Vector Additivity: s * (m1 + m2) = s*m1 + s*m2
2. Scalar Additivity: (s1 + s2) * m = s1*m + s2*m
3. Associativity of Scalar Multiplication: (s1 * s2) * m = s1 * (s2 * m)
4. Identity of Scalar Multiplication: 1 * m = m

We verify these axioms statically where possible.

For IsMatrixScalar F, this verifies:

• F satisfies the operational scalar contract used by the geometry layer
• LinearMap supports the external linear action
• Addition and scalar multiplication are properly closed