🧮 格拉姆-施密特正交化计算器Gram–Schmidt Calculator

使用说明与常见问题How-to & FAQ

「格拉姆-施密特正交化计算器」是什么？What is the Gram–Schmidt Calculator tool?

输入一组向量，用格拉姆-施密特过程精确求出正交基（分数）和标准正交基（小数，并给出每个向量的长度），线性相关的向量会被识别并跳过。Enter a set of vectors to get an orthogonal basis by the Gram–Schmidt process (exact fractions) and an orthonormal basis (decimals, with each vector's length); linearly dependent vectors are detected and skipped.

如何使用「格拉姆-施密特正交化计算器」？How do I use the Gram–Schmidt Calculator tool?

在「格拉姆-施密特正交化计算器」中，填写「小数位数」等参数，结果会随输入实时计算并显示在下方。In the Gram–Schmidt Calculator: Fill in "Decimal places"; results are recalculated instantly as you type.

格拉姆-施密特过程是什么？What is the Gram–Schmidt process?

把一组线性无关的向量逐个变成两两正交的向量：每个新向量减去它在前面各正交向量上的投影，uₖ = vₖ − Σ proj_{uⱼ}(vₖ)。再把每个 uₖ 除以自身长度就得到标准正交基。It turns linearly independent vectors into mutually orthogonal ones: subtract from each vector its projections onto the earlier orthogonal vectors, uₖ = vₖ − Σ proj_{uⱼ}(vₖ). Dividing each uₖ by its length gives an orthonormal basis.

线性相关的向量怎么处理？What happens to dependent vectors?

如果某个向量减去投影后变成零向量，说明它能由前面的向量线性表示，会被跳过并在结果中列出，正交基的个数就是这组向量张成空间的维数。If a vector becomes zero after subtracting the projections, it is a combination of the earlier ones; it is skipped and listed, and the number of basis vectors is the dimension of the span.

计算精度如何？How precise are the results?

正交向量 uₖ 和 ‖uₖ‖² 用分数（BigInt）精确计算，没有舍入误差；标准正交向量 eₖ 需要开平方，用 JavaScript 双精度浮点数（约 15–16 位有效数字）计算，再按所选小数位显示。The orthogonal vectors uₖ and ‖uₖ‖² are computed exactly with fractions (BigInt), with no rounding; the orthonormal vectors eₖ need square roots, so they use JavaScript double-precision numbers (about 15–16 significant digits) and are shown to the chosen decimal places.

「格拉姆-施密特正交化计算器」免费吗？需要注册吗？Is the Gram–Schmidt Calculator free? Do I need an account?

「格拉姆-施密特正交化计算器」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 Gram–Schmidt Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.