⟂ 向量线性无关判定器（任意个数和维数，含秩与相关关系）Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency)
每行输入一个向量（分量用逗号或空格分隔，最多 8 个向量、8 维），用高斯消元求矩阵的秩，判断这些向量是否线性无关；如果线性相关，给出一组使线性组合为零向量的系数。二维情形同时显示行列式。本地处理。Enter one vector per line (components separated by commas or spaces, up to 8 vectors of up to 8 components) to get the rank by Gaussian elimination and decide whether the vectors are linearly independent; if they are dependent, a set of coefficients that combines them to the zero vector is given. For two 2D vectors the determinant is shown too. Processed locally.
使用说明与常见问题How-to & FAQ
「向量线性无关判定器（任意个数和维数，含秩与相关关系）」是什么？What is the Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency) tool?
每行输入一个向量（分量用逗号或空格分隔，最多 8 个向量、8 维），用高斯消元求矩阵的秩，判断这些向量是否线性无关；如果线性相关，给出一组使线性组合为零向量的系数。二维情形同时显示行列式。本地处理。Enter one vector per line (components separated by commas or spaces, up to 8 vectors of up to 8 components) to get the rank by Gaussian elimination and decide whether the vectors are linearly independent; if they are dependent, a set of coefficients that combines them to the zero vector is given. For two 2D vectors the determinant is shown too. Processed locally.
如何使用「向量线性无关判定器（任意个数和维数，含秩与相关关系）」？How do I use the Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency) tool?
在「向量线性无关判定器（任意个数和维数，含秩与相关关系）」中，填写上方字段等参数，结果会随输入实时计算并显示在下方。In the Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency): Fill in the fields above; results are recalculated instantly as you type.
线性无关是什么意思？What does linearly independent mean?
如果只有全部系数都为 0 时 c₁v₁ + c₂v₂ + … + cₖvₖ 才等于零向量，这组向量就线性无关；否则某个向量可以由其他向量组合得到，称为线性相关。Vectors are independent if c₁v₁ + c₂v₂ + … + cₖvₖ = 0 only when every coefficient is 0. Otherwise one vector can be built from the others and they are dependent.
怎么判断？How is it decided?
把向量作为矩阵的列做高斯消元，秩等于向量个数时线性无关。方阵时也可以看行列式是否为 0。Put the vectors in a matrix as columns and row-reduce; they are independent when the rank equals the number of vectors. For a square matrix, a non-zero determinant says the same.
相关关系怎么读？How do I read the dependency?
例如 2·v1 − 1·v2 = 0 表示 v2 = 2·v1。系数不唯一，任意非零倍数也成立。For example 2·v1 − 1·v2 = 0 means v2 = 2·v1. The coefficients are not unique; any non-zero multiple also works.
能举个计算实例吗？Can you show a worked example?
例如在「向量线性无关判定器（任意个数和维数，含秩与相关关系）」中填 向量（每行一个） = 1, 0（换行）0, 1，结果区显示：秩：2；结论：线性无关；行列式：1For example, entering Vectors (one per line) = 1, 0 (new line) 0, 1 in the Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency) gives: Rank: 2; Result: Linearly independent; Determinant: 1
计算精度如何？How precise are the results?
「向量线性无关判定器（任意个数和维数，含秩与相关关系）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「向量线性无关判定器（任意个数和维数，含秩与相关关系）」免费吗？需要注册吗？Is the Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency) free? Do I need an account?
「向量线性无关判定器（任意个数和维数，含秩与相关关系）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Linear Independence Calculator (Any Number and Size of Vectors, with Rank and Dependency) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.