From 0e6bfc9a79525831d705b3cb6eb8a7596180eacb Mon Sep 17 00:00:00 2001 From: pjokerx <1433560268@qq.com> Date: Sat, 17 Jan 2026 14:39:27 +0800 Subject: [PATCH 1/6] vault backup: 2026-01-17 14:39:26 --- 素材/正交矩阵和施密特正交化法.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/素材/正交矩阵和施密特正交化法.md b/素材/正交矩阵和施密特正交化法.md index d386e9d..bc861a4 100644 --- a/素材/正交矩阵和施密特正交化法.md +++ b/素材/正交矩阵和施密特正交化法.md @@ -75,7 +75,7 @@ $$\boldsymbol A^{-1} = \frac{1}{|A|}\boldsymbol{A}^*$$ $$\boldsymbol{A}^T= \frac{1}{|A|}\boldsymbol A^*$$ 正交矩阵的行列式满足 $\frac{1}{|A|} =±1$，故 $A^*={|A|}A^T=±A^T$ -由伴随矩阵的定义，其第 (j,i)元为 aij​的代数余子式 Aij​，而 ±AT的第 (j,i)元为 ±aij​。比较对应元素得 +由伴随矩阵的定义，其第 (j,i)元为 $a_{ij}$​的代数余子式 $A_{ij}$​，而 $±A^T$的第 (j,i)元为 $±a_{ij}$​。比较对应元素得 $$A_{ij}​=±a_{ij}​,i,j=1,2,…,n.$$ 证毕 From b40664fa0ccbbbfda60c5e7520dda715d376703b Mon Sep 17 00:00:00 2001 From: unknown <18951088369@163.com> Date: Sat, 17 Jan 2026 14:41:03 +0800 Subject: [PATCH 2/6] vault backup: 2026-01-17 14:41:03 --- 素材/正交矩阵和施密特正交化法.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/素材/正交矩阵和施密特正交化法.md b/素材/正交矩阵和施密特正交化法.md index d386e9d..dcae4a6 100644 --- a/素材/正交矩阵和施密特正交化法.md +++ b/素材/正交矩阵和施密特正交化法.md @@ -99,12 +99,12 @@ k=2,3,\dots,p$$ ### **例子** >[!example] **例1** -已知 为欧氏空间 V 的一组标准正交基，令$\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\dots,\boldsymbol{\alpha}_5$ $$\boldsymbol{\beta}_1 = \boldsymbol{\alpha}_1+\boldsymbol{\alpha}_3,\quad +已知 $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\dots,\boldsymbol{\alpha}_5$ 为欧氏空间 V 的一组标准正交基，令$$\boldsymbol{\beta}_1 = \boldsymbol{\alpha}_1+\boldsymbol{\alpha}_3,\quad \boldsymbol{\beta}_2 = \boldsymbol{\alpha}_1-\boldsymbol{\alpha}_2+\boldsymbol{\alpha}_4,\quad \boldsymbol{\beta}_3 = 2\boldsymbol{\alpha}_1+\boldsymbol{\alpha}_2+\boldsymbol{\alpha}_3,$$ $U = \text{span}\{\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\boldsymbol{\beta}_3\}$求 U 的一个标准正交基。 -**解析**。 +**解析**： 施密特正交化 步骤1：正交化 取$$ \boldsymbol{\gamma}_1=\boldsymbol{\beta}_1=\boldsymbol{\alpha}_1+\boldsymbol{\alpha}_3 From 0999b52b2ef053b83a2939a53355011259516a04 Mon Sep 17 00:00:00 2001 From: pjokerx <1433560268@qq.com> Date: Sat, 17 Jan 2026 14:42:43 +0800 Subject: [PATCH 3/6] vault backup: 2026-01-17 14:42:43 --- 素材/正交矩阵和施密特正交化法.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/素材/正交矩阵和施密特正交化法.md b/素材/正交矩阵和施密特正交化法.md index 29c5816..86f636b 100644 --- a/素材/正交矩阵和施密特正交化法.md +++ b/素材/正交矩阵和施密特正交化法.md @@ -93,9 +93,8 @@ $$\begin{align*} \boldsymbol{u}_k &= \boldsymbol{\alpha}_k - \sum_{i=1}^{k-1}\frac{\langle\boldsymbol{\alpha}_k,\boldsymbol{u}_i\rangle}{\langle\boldsymbol{u}_i,\boldsymbol{u}_i\rangle}\boldsymbol{u}_i,\quad k=2,3,\dots,p. \end{align*}$$ 单位化过程 -$$\boldsymbol{\varepsilon}_1 = \frac{\boldsymbol{\alpha}_1}{\|\boldsymbol{\alpha}_1\|}$$ $$\boldsymbol{\varepsilon}_k = \frac{\boldsymbol{u}_k}{\|\boldsymbol{u}_k\|},\quad -k=2,3,\dots,p$$ +k=1,2,3,\dots,p$$ ### **例子** >[!example] **例1** From 6d4684ff1e4f9cbb689f9f529d78d0efd5155009 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=E5=88=98=E6=9F=AF=E5=A6=A4?= <2503393720@qq.com> Date: Sat, 17 Jan 2026 14:43:46 +0800 Subject: [PATCH 4/6] vault backup: 2026-01-17 14:43:46 --- 素材/正交矩阵和施密特正交化法.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/素材/正交矩阵和施密特正交化法.md b/素材/正交矩阵和施密特正交化法.md index d386e9d..2e81e45 100644 --- a/素材/正交矩阵和施密特正交化法.md +++ b/素材/正交矩阵和施密特正交化法.md @@ -142,7 +142,7 @@ $$\begin{cases} \langle\boldsymbol{x},\boldsymbol{\alpha}_3\rangle = x_1 - 2x_2 + 2x_4 = 0\\ \langle\boldsymbol{x},\boldsymbol{\alpha}_4\rangle = 2\sqrt{6}x_1 - \sqrt{6}x_3 - \sqrt{6}x_4 = 0 \implies 2x_1 - x_3 - x_4 = 0 \end{cases}$$ -解上述齐次方程组，基础解系，得到两个线性无关的解： +解上述齐次方程组,得到两个线性无关的解： $$\boldsymbol{\xi}_1=(2,1,4,0)^T,\quad \boldsymbol{\xi}_2=(0,1,0,1)^T$$ 正交化 From ba89c7a6c7a3b73b34c25a8f4c508aa0a4b8f3a3 Mon Sep 17 00:00:00 2001 From: pjokerx <1433560268@qq.com> Date: Sat, 17 Jan 2026 14:48:12 +0800 Subject: [PATCH 5/6] vault backup: 2026-01-17 14:48:12 --- 素材/正交矩阵和施密特正交化法.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/素材/正交矩阵和施密特正交化法.md b/素材/正交矩阵和施密特正交化法.md index 86f636b..4e0e6d4 100644 --- a/素材/正交矩阵和施密特正交化法.md +++ b/素材/正交矩阵和施密特正交化法.md @@ -106,7 +106,7 @@ $U = \text{span}\{\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\boldsymbol{\beta}_3 **解析**： 施密特正交化 步骤1：正交化 -取$$ \boldsymbol{\gamma}_1=\boldsymbol{\beta}_1=\boldsymbol{\alpha}_1+\boldsymbol{\alpha}_3 +取$$ \boldsymbol{\gamma}_1=\boldsymbol{\beta}_1=\boldsymbol{\alpha}_1+\boldsymbol{\alpha}_3,\ \ \boldsymbol{\gamma}_2=\boldsymbol{\beta}_2-\dfrac{\langle\boldsymbol{\beta}_2,\boldsymbol{\gamma}_1\rangle}{\langle\boldsymbol{\gamma}_1,\boldsymbol{\gamma}_1\rangle}\boldsymbol{\gamma}_1$$$$\langle\boldsymbol{\beta}_2,\boldsymbol{\gamma}_1\rangle=1,\quad \langle\boldsymbol{\gamma}_1,\boldsymbol{\gamma}_1\rangle=2$$ $$\boldsymbol{\gamma}_2=\frac{1}{2}\boldsymbol{\alpha}_1-\boldsymbol{\alpha}_2-\frac{1}{2}\boldsymbol{\alpha}_3+\boldsymbol{\alpha}_4$$ From 1558431cef9922e18ae66d6c52ee6ecf7fbf4e77 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=E7=8E=8B=E8=BD=B2=E6=A5=A0?= Date: Sat, 17 Jan 2026 14:52:32 +0800 Subject: [PATCH 6/6] vault backup: 2026-01-17 14:52:32 --- 编写小组/讲义/微分中值定理.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/编写小组/讲义/微分中值定理.md b/编写小组/讲义/微分中值定理.md index 67845bc..e32e860 100644 --- a/编写小组/讲义/微分中值定理.md +++ b/编写小组/讲义/微分中值定理.md @@ -140,7 +140,7 @@ $$ ``` >[!example] 例3 -设 $f(x)$ 在 $[0, 1]$ 上可导，且$f(1) = 2\sqrt{e}\int_0^{1/2} e^{1-x} f(x) dx$ +设 $f(x)$ 在 $[0, 1]$ 上可导，且$f(1) = 2\sqrt{e}\int_0^{1/2} e^{\frac{x^2}{2}-x} f(x) dx$ 证明：存在 $\xi \in (0, 1)$ 使得：$f'(\xi) = (1-\xi) f(\xi)$ ```text