Abstract Vector Spaces & Linear Maps

Same algebra, new objects

Polynomials, signal spaces, and function spaces obey the same axioms as ℝⁿ. A linear map T satisfies T(u+v)=T(u)+T(v) and T(cu)=cT(u).

• Kernel and image generalise null space and column space.
• Matrix of T depends on choice of bases — change-of-basis matrices conjugate representations.
• Isomorphisms preserve dimension and linear structure.