전체 조망

Current Understanding

The wiki follows Serge Lang's Linear Algebra (3rd ed), chosen for its theoretical rigor: everything is developed abstractly over an arbitrary field $K$, results are proved in full generality, and linear maps are the primary objects (matrices are their coordinate representations). See 읽기 경로 for the chapter-by-chapter route and 원전 소개 — Lang for the source.

The conceptual spine

1. Structure first (체와 벡터공간 → 기저와 차원 → 합과 직합): a vector space is axioms VS 1–8 over a field; dimension is well defined via the exchange theorem; spaces decompose as direct sums of subspaces.
2. Maps, then matrices (선형사상 → 선형사상과 행렬): rank–nullity governs every linear map; the map↔matrix correspondence is an isomorphism, and changing basis conjugates the matrix — similarity is the central equivalence.
3. Geometry from a scalar product (스칼라곱과 직교성 → 쌍대공간과 쌍선형형식): orthonormal bases, the orthogonal complement, $V\cong V^*$, and Sylvester's signature.
4. The determinant (행렬식): the unique multilinear alternating normalized form; multiplicative, the obstruction to invertibility, the source of the characteristic polynomial.
5. Spectral theory (대칭·에르미트·유니터리 연산자 → 고윳값과 특성다항식 → 스펙트럼 정리): nice operators get an orthonormal eigenbasis.
6. Canonical forms via $K[t]$ (연산자의 다항식 → 삼각화와 일차분해 → 조르당 표준형): Cayley–Hamilton, primary decomposition, and Jordan form — essentially the structure theorem for $\mathbb{C}[t]$-modules applied to one operator.
7. Outward bridge (볼록집합과 크레인–밀만 정리): convexity, separating/supporting hyperplanes, Krein–Milman — toward optimization and functional analysis.

The recurring engine throughout is the pair (direct-sum decomposition into invariant subspaces, non-degenerate scalar product) — the first yields canonical forms, the second yields adjoints and the spectral theorem.

Open Questions

• Lang's $K[t]$-module machinery (Ch IX–XI) connects directly to abstract algebra; if an Algebra wiki (Hungerford/Lang) is later built, primary decomposition and Jordan form are natural cross-links to the structure theorem for modules over a PID.
• A computational companion (row reduction, LU/QR, numerical methods) is intentionally outside Lang's scope; could be added from a Strang/Anton source if desired.

Key Entities / Concepts

• Foundations: 체와 벡터공간, 기저와 차원, 합과 직합
• Maps & matrices: 행렬과 선형방정식, 선형사상, 선형사상과 행렬
• Products & duality: 스칼라곱과 직교성, 쌍대공간과 쌍선형형식
• Determinants: 행렬식
• Spectral theory: 대칭·에르미트·유니터리 연산자, 고윳값과 특성다항식, 스펙트럼 정리
• Canonical forms: 연산자의 다항식, 삼각화와 일차분해, 조르당 표준형
• Advanced: 볼록집합과 크레인–밀만 정리
• Reading path: 읽기 경로