Fun with polynomials and linear algebra; or, slight abstract nonsense
This technical dive explores how fundamental linear algebra concepts, like vector spaces and quotients, can elegantly unify and generalize theorems often found in abstract algebra and number theory. By applying these structures to polynomials, the author re-derives and simplifies results like the Chinese Remainder Theorem. It appeals to HN's audience interested in deep mathematical connections and foundational proofs.
The Lowdown
This post, described as a series of notes for the author, endeavors to demonstrate the power and generality of linear algebra by re-framing and generalizing various mathematical theorems and constructions using purely linear-algebraic language. The author shows how many concepts, often associated with module theory or abstract algebra, can be understood and proven within the framework of vector spaces and their properties.
- Basic Linear Algebra Foundations: The article begins by reviewing core definitions such as vector space isomorphism (V ≃ V'), injectivity, kernels, and the properties of finite-dimensional subspaces and their dimensions. It highlights that if two finite-dimensional spaces U and W have the same dimension, U ≃ W. A key observation is that an injective or surjective linear map between same-dimension finite-dimensional spaces is invertible.
- Quotients of Vector Spaces: The concept of V/W, a quotient space formed by a vector space V and its subspace W, is introduced. This is analogous to cosets in group theory but applied to vector spaces. The article proves that V/W is itself a vector space and, importantly, establishes a structure theorem: if U and W are subspaces of V with dim(U) = dim(V/W) and U ∩ W = {0}, then V can be uniquely decomposed as V = U ⊕ W.
- Polynomial Fun: These linear-algebraic tools are then applied to the vector space of polynomials. The set of polynomials divisible by a polynomial 'p' (denoted Wp) is shown to be a subspace. The author then demonstrates how operations like Wp + Wq relate to the greatest common divisor (gcd) of p and q, and Wp ∩ Wq to their least common multiple (lcm). A central result is showing that V/Wp is isomorphic to Rp, the set of polynomials with degree less than deg(p), leading to the decomposition V = Wp ⊕ Rp.
- Generalized Chinese Remainder Theorem: The article culminates in a generalized version of the Chinese Remainder Theorem. It states that if W1, ..., Wn are subspaces of V such that V/⋂iWi is finite-dimensional, then V/⋂iWi ≃ ∏i(V/Wi) if and only if dim(V/⋂iWi) = ∑idim(V/Wi). This condition is then explicitly applied to polynomials: for mutually coprime polynomials p1, ..., pn, the degrees align, thereby showing that the familiar polynomial Chinese Remainder Theorem is a specific instance of this general linear-algebraic statement. The canonical map and its inverse are detailed, showing how polynomial decomposition works in this framework.
- The Basis View: For finite-dimensional V, the decomposition is briefly interpreted using stacked matrices, where the condition ⋂iWi = {0} corresponds to a stacked matrix having a trivial kernel, implying invertibility.
Overall, the post provides a compelling argument for the unifying power of linear algebra, illustrating how abstract concepts can simplify and illuminate connections between different mathematical domains, making complex ideas more accessible and elegant.