線形代数において、ランク・ヌル性定理は、有限次元ベクトル空間間の任意の線形写像[latex]T: V to W[/latex]について、その定義域[latex]V[/latex]の次元は、ランク（像の次元）とヌル性（核の次元）の和に等しいことを述べている。式は[latex]dim(V) = text{rank}(T) + text{nullity}(T)[/latex]である。
ランク・ヌルティ定理
1884
- James Joseph Sylvester
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The rank-nullity theorem provides a fundamental relationship between the dimensions of the key subspaces associated with a linear transformation. Let [latex]T: V to W[/latex] be a linear map. The kernel of T, denoted [latex]ker(T)[/latex], is the set of vectors in [latex]V[/latex] that are mapped to the zero vector in [latex]W[/latex]. The dimension of the kernel is called the nullity of T. The image of T, denoted [latex]text{im}(T)[/latex], is the set of all vectors in [latex]W[/latex] that are the output of T for some input vector from [latex]V[/latex]. The dimension of the image is the rank of T.
The theorem states [latex]dim(text{domain}(T)) = dim(ker(T)) + dim(text{im}(T))[/latex]. A common proof strategy involves constructing a basis. First, find a basis for the kernel, say [latex]{u_1, dots, u_k}[/latex], where [latex]k = text{nullity}(T)[/latex]. Since the kernel is a subspace of [latex]V[/latex], this basis can be extended to a basis for all of [latex]V[/latex]: [latex]{u_1, dots, u_k, v_1, dots, v_r}[/latex]. The dimension of [latex]V[/latex] is thus [latex]k+r[/latex]. The final step is to show that the set [latex]{T(v_1), dots, T(v_r)}[/latex] forms a basis for the image of T. This proves that the rank is [latex]r[/latex], and therefore [latex]dim(V) = k+r = text{nullity}(T) + text{rank}(T)[/latex].
For matrices, if [latex]A[/latex] is an [latex]m times n[/latex] matrix, it represents a linear map from [latex]mathbb{R}^n[/latex] to [latex]mathbb{R}^m[/latex]. The domain’s dimension is [latex]n[/latex]. The rank of [latex]A[/latex] is the dimension of its column space, and its nullity is the dimension of its null space. The theorem becomes [latex]n = text{rank}(A) + text{nullity}(A)[/latex].
The theorem is a core component of what is sometimes called the fundamental theorem of linear algebra, which provides a comprehensive description of the structure of the four fundamental subspaces associated with an [latex]m times n[/latex] matrix [latex]A[/latex]: the column space, the null space, the row space, and the left null space. It beautifully illustrates the trade-off that as the set of solutions to [latex]Ax=0[/latex] (the null space) grows larger, the set of possible outputs [latex]Ax[/latex] (the column space) must become smaller, with their dimensions summing to the total dimension of the input space.
UNESCO Nomenclature: 1201
代数
タイプ
抽象システム
混乱
基礎
使用法
広く普及している
前駆物質
- concept of a vector space, formalized by giuseppe peano.
- theory of matrices, developed by arthur cayley.
- concept of dimension of a vector space, developed by hermann grassmann.
- understanding of linear independence and basis vectors.
- formalization of linear transformations (maps) between vector spaces.
- early work on solving systems of linear equations by carl friedrich gauss.
アプリケーション
- solving systems of linear equations
- principal component analysis (pca) in data science
- computer graphics transformations
- control theory in engineering
- error detection and correction codes
- 暗号学
- quantum mechanics
特許:
NA
潜在的なイノベーションのアイデア
- optimizing ニューラルネットワーク architectures by analyzing the rank and nullity of weight matrices to prune redundant neurons.
- developing more efficient data compression algorithms by identifying and exploiting the null space of data transformation matrices.
- creating novel error-correcting codes based on the properties of the kernel of a linear map representing the encoding process.
- enhancing quantum computing algorithms by using the rank-nullity theorem to analyze the state space of quantum systems under linear transformations.
- designing advanced control systems for robotics where the null space of a jacobian matrix represents redundant motions that can be used for obstacle avoidance.
- improving recommendation systems by analyzing the rank of user-item matrices to understand the dimensionality of user preferences.
- developing new cryptographic techniques where the security relies on the difficulty of finding the kernel of a large, sparse linear transformation.
- creating generative art algorithms that use the kernel of a transformation to generate variations within a defined stylistic space.
- analyzing financial models by applying the theorem to matrices of economic indicators to identify independent factors and dependencies.
- で bioinformatics, analyzing gene expression data matrices to determine the effective number of independent regulatory pathways (rank) and co-regulated gene sets (null space).
Related to: rank-nullity theorem, linear algebra, vector space, dimension, kernel, nullity, image, rank, linear transformation, matrix theory.
歴史的背景
ランク・ヌルティ定理
（日付が不明または関連性がない場合、例えば「流体力学」などでは、その注目すべき出現時期の概算値が提示されます。）
