V2EX linear map

Linear Map

Definition / 释义

linear map线性映射(线性变换)。指在向量空间之间满足

  1. \(T(u+v)=T(u)+T(v)\)(加法保持)
  2. \(T(cv)=cT(v)\)(数乘保持)
    的函数/映射。常可用矩阵表示。

Pronunciation / 发音(IPA)

/lnir mp/

Examples / 例句

A linear map sends the zero vector to the zero vector.
线性映射会把零向量映到零向量。

If \(T\) is a linear map from \(V\) to \(W\), then \(T\) is completely determined by its values on a basis of \(V\).
如果 \(T\) 是从 \(V\) 到 \(W\) 的线性映射,那么 \(T\) 在 \(V\) 的一组基上的取值就能完全决定 \(T\)。

Etymology / 词源

linear 来自拉丁语 linearis(“线的、直线的”),与 linea(“线”)有关;map 源自中世纪拉丁语 mappa(“布、地图”),后引申为“映射/对应关系”。在数学里,linear map 用来强调这种“保持线性结构”的对应关系。

Related Words / 相关词汇

Notable Works / 典籍与著作

  • Linear Algebra Done Right(Sheldon Axler)
  • Introduction to Linear Algebra(Gilbert Strang)
  • Finite-Dimensional Vector Spaces(Paul R. Halmos)
  • Linear Algebra(Serge Lang)
  • Algebra(Michael Artin)
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