你能找到一个与 N 个向量都不正交的向量吗?
本文探讨了一个有趣的线性代数问题:给定 N 个非零向量,是否总能找到一个向量与它们每一个都不正交?作者从二维和三维空间的直观例子出发,逐步推广到高维情况,并给出了构造性证明。文章还讨论了该问题的实际应用背景,以及相关的几何直觉。
本文探讨了一个有趣的线性代数问题:给定 N 个非零向量,是否总能找到一个向量与它们每一个都不正交?作者从二维和三维空间的直观例子出发,逐步推广到高维情况,并给出了构造性证明。文章还讨论了该问题的实际应用背景,以及相关的几何直觉。
A blog post discusses a mathematical identity where pentagonal numbers can be expressed in terms of triangular numbers. It highlights that while examples don't typically prove theorems, in this case the identity Pn = T(2n−1) − T(n−1) holds, showing that three examples can suffice for proving certain relationships.
John D. Cook describes how a sequence of his blog posts often follows a hidden thread, beginning with a post about the mathematical approximation exp(−x²) ≈ (1 + cos(sin(x) + x))/2, which some commenters incorrectly attributed solely to a first-order Taylor expansion.
The nth pentagonal number Pn follows the formula Pn = (3n² − n)/2 for positive integer n. For non-positive integer n, the same formula defines a generalized pentagonal number.
Partial fraction decomposition is commonly introduced in calculus as a technique for integrating rational functions by breaking P(x)/Q(x) into simpler terms. However, the post suggests that this method has applications beyond integration that are often overlooked in a typical calculus class.