From 9b3466b8bf9847319d1bf7912c54dfd1a602699a Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?=E5=88=98=E6=9F=AF=E5=A6=A4?= <2503393720@qq.com>
Date: Sun, 18 Jan 2026 17:39:20 +0800
Subject: [PATCH] vault backup: 2026-01-18 17:39:20

---
 素材/未命名.md | 8 ++++++++
 1 file changed, 8 insertions(+)
 create mode 100644 素材/未命名.md

diff --git a/素材/未命名.md b/素材/未命名.md
new file mode 100644
index 0000000..c69749f
--- /dev/null
+++ b/素材/未命名.md
@@ -0,0 +1,8 @@
+设  $\boldsymbol{\beta}_1, \boldsymbol{\beta}_2, \dots, \boldsymbol{\beta}_m$  均为实数域上的  n  维列向量，其中  m < n ，证明  $\boldsymbol{\beta}_1, \boldsymbol{\beta}_2, \dots, \boldsymbol{\beta}_m$  线性无关的充要条件为
+$$\begin{vmatrix}
+\boldsymbol{\beta}_1^\mathrm{T}\boldsymbol{\beta}_1 & \boldsymbol{\beta}_1^\mathrm{T}\boldsymbol{\beta}_2 & \dots & \boldsymbol{\beta}_1^\mathrm{T}\boldsymbol{\beta}_m \\
+\boldsymbol{\beta}_2^\mathrm{T}\boldsymbol{\beta}_1 & \boldsymbol{\beta}_2^\mathrm{T}\boldsymbol{\beta}_2 & \dots & \boldsymbol{\beta}_2^\mathrm{T}\boldsymbol{\beta}_m \\
+\vdots & \vdots & & \vdots \\
+\boldsymbol{\beta}_m^\mathrm{T}\boldsymbol{\beta}_1 & \boldsymbol{\beta}_m^\mathrm{T}\boldsymbol{\beta}_2 & \dots & \boldsymbol{\beta}_m^\mathrm{T}\boldsymbol{\beta}_m
+\end{vmatrix}
+\neq 0.$$