rank–nullity theorem

维数公式

Algebra & Algebraic Geometry · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a linear map T:V→WT:V\to W with VV finite-dimensional, dim⁡V=dim⁡ker⁡T+dim⁡im⁡T\dim V=\dim\ker T+\dim\operatorname{im}T.

有限维定义域上的线性映射满足定义域维数等于核的维数与像的维数之和。

Example

Differentiation on F[x]≤2F[x]_{\le2} in characteristic zero has nullity one and rank two.

Usage and distinctions

The finite-dimensional form uses ordinary integer addition; subtraction of infinite cardinals is not an equivalent formulation.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Wen-Wei Li — Elementary Algebra lecture notes, author manuscript§4.8 核, 像与消元法, Theorem 4.8.4, finite-dimensional domain hypothesis and kernel/image dimension formula, printed p.146 / PDF p.156 · Checked 2026-10-15

Translation sources

  1. 厦门大学数学科学学院 — 高等代数教案（08版）§4.4 线性映射的像与核, 推论3（线性映射的维数公式）及紧接的证明, PDF pp.2–3 · Checked 2026-10-15
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