linear independence

线性无关

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

Translation evidence: Literature usage

Meaning in context

A family (vi)(v_i) is linearly independent if every finite relation ∑iaivi=0\sum_i a_iv_i=0 has all coefficients zero.

向量族的任何有限线性组合等于零时，各系数都必须为零。

Example

(1,0),(0,1)(1,0),(0,1) are independent over R\mathbb R.

Usage and distinctions

Only finite sums occur in algebraic linear independence, even for infinite families.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Wen-Wei Li — Methods in Algebra I, author source自由模 (sec:free-modules), definition immediately after prop:free-module-property, first item 线性无关 and following finite-relation equation; specialize R to a field · Checked 2026-10-15

Translation sources

  1. 李文威 — 代数学方法 卷一，作者源文件自由模 (sec:free-modules), definition immediately after prop:free-module-property, first item 线性无关 and following finite-relation equation; specialize R to a field · Checked 2026-10-15
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