From 18da8b2f677bfe00525d54dca47bd1dbaf82650c Mon Sep 17 00:00:00 2001 From: Elwood <3286545699@qq.com> Date: Fri, 2 Jan 2026 01:10:35 +0800 Subject: [PATCH 1/5] vault backup: 2026-01-02 01:10:35 --- 编写小组/试卷/1231线性代数考试卷(解析版).md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/编写小组/试卷/1231线性代数考试卷(解析版).md b/编写小组/试卷/1231线性代数考试卷(解析版).md index aa594eb..f9048b1 100644 --- a/编写小组/试卷/1231线性代数考试卷(解析版).md +++ b/编写小组/试卷/1231线性代数考试卷(解析版).md @@ -45,7 +45,7 @@ tags: >C. $\begin{bmatrix}\alpha_1&\alpha_3&\alpha_4\end{bmatrix}=\begin{bmatrix}0&1&-1\\0&-1&1\\c_1&c_3&c_4\end{bmatrix}\cong\begin{bmatrix}0&0&-1\\0&0&1\\c_1&c_3+c_4&c_4\end{bmatrix}\cong\begin{bmatrix}0&0&0\\0&0&1\\c_1&c_3+c_4&c_4\end{bmatrix}$,故其必定线性相关; >D. $\begin{bmatrix}\alpha_1&\alpha_2&\alpha_4\end{bmatrix}=\begin{bmatrix}0&1&-1\\1&-1&1\\c_2&c_3&c_4\end{bmatrix}\cong\begin{bmatrix}0&0&-1\\1&0&1\\c_2&c_3+c_4&c_4\end{bmatrix}$,当$c_3+c_4\ne 0$时,作初等列变换,$\text{rank}\begin{bmatrix}0&0&-1\\1&0&1\\c_2&c_3+c_4&c_4\end{bmatrix}=\text{rank}\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}=3$,故线性无关; -4. 设 $A, B$ 为 $n$ 阶矩阵,则 +4. 设 $A, B$ 为 $n$ 阶矩阵,则 (A) $\text{rank}[A \ AB] = \text{rank} A$; (B) $\text{rank}[A \ BA] = \text{rank} A$; (C) $\text{rank}[A \ B] = \max\{\text{rank} A, \text{rank} B\}$; From 21185704ce19d0a56608945ad6cb8e039b197f1d Mon Sep 17 00:00:00 2001 From: Elwood <3286545699@qq.com> Date: Sat, 3 Jan 2026 16:17:09 +0800 Subject: [PATCH 2/5] vault backup: 2026-01-03 16:17:09 --- 编写小组/试卷/0103高数模拟试卷(解析版).md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/编写小组/试卷/0103高数模拟试卷(解析版).md b/编写小组/试卷/0103高数模拟试卷(解析版).md index 5a3ea54..2616d35 100644 --- a/编写小组/试卷/0103高数模拟试卷(解析版).md +++ b/编写小组/试卷/0103高数模拟试卷(解析版).md @@ -398,7 +398,7 @@ $$ **解:** -当 $x \to 0$ 时,利用等价无穷小替换和泰勒展开: +当 $x \to 0$ 时,利用等价无穷小替换和泰勒展开:9 分母:$x^3 \arcsin x \sim x^3 \cdot x = x^4$ From 7d2d7d0662f7ef95da36525b885f03de2fb2c018 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=E7=8E=8B=E8=BD=B2=E6=A5=A0?= Date: Fri, 9 Jan 2026 00:03:23 +0800 Subject: [PATCH 3/5] =?UTF-8?q?=E7=A0=94=E8=AE=A8=E8=AE=B0=E5=BD=952026.1.?= =?UTF-8?q?8=E6=99=9A?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 研讨记录/2026.1.8.md | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) create mode 100644 研讨记录/2026.1.8.md diff --git a/研讨记录/2026.1.8.md b/研讨记录/2026.1.8.md new file mode 100644 index 0000000..7f8316a --- /dev/null +++ b/研讨记录/2026.1.8.md @@ -0,0 +1,23 @@ +平时进行雨课堂等上题目的交流 +打电话交流?no +之后自习会增多,我们可以争取一些自习课来,这个后面再说。体能。 +集中—>解答困惑,素材 +临时部署 +雨课堂和线代作业的整理,加班时间可能会增加 +**作业上找题目(平时小测)** 2~3道题目 +每两天进行一次研讨小会(一三五日,晚或体能)完成作业,分享思路——强迫比较好地完成作业。选择小测题目。复习。 +陈搜集资料(教员习题课),王嘉兴 +线代:真题优先,习题册(关键在于没有答案),期末考和习题册是有很大重叠的 +**提前确定会议主题**(如)周三:线代作业 +讨论主题:主要不是讨论答案是什么,而是使用的方法的适用条件,让一类题的做法形成一套’机械化‘的流程(过于理想化了) +**每周一套/两套试卷**,在什么量的情况下是可以接受而且必须要给自己逼一把的;灵活处理,不需要固定在哪天写哪些题。 +**确保作业全对。** 用py计算线性代数来检查。 +原则:**把大的目标落实到每一次研讨中** +**灵活处理,下次之前决定下次的议题** +确保出的题目得是自己做过的 +**讲义和学习资料** +安排几个平时笔记做得好的去跟线代习题课 + +分工问题。研讨中出试卷、讲义分工问题的解决。挑题、讨论、成文。 + +下次开会时间:1.11周日晚 \ No newline at end of file From 829cc1260feeade648acf9c0f653a3c1d06f43d8 Mon Sep 17 00:00:00 2001 From: Cym10x Date: Fri, 9 Jan 2026 00:03:52 +0800 Subject: [PATCH 4/5] Modify README --- README.md | 19 +++++++++++++++++-- 素材/图片/README/edit1.png | Bin 0 -> 397820 bytes 2 files changed, 17 insertions(+), 2 deletions(-) create mode 100644 素材/图片/README/edit1.png diff --git a/README.md b/README.md index 33c9ffe..390c1bd 100644 --- a/README.md +++ b/README.md @@ -1,2 +1,17 @@ -# 一起数学----高等数学笔记协作平台 -大家可以把Markdown高数笔记上传到此处 \ No newline at end of file +# 讲义编写研讨组欢迎你! +### 开始工作前的准备: +##### 概述 +1. 下载Obsidian和Git For Windows(如果官网下不来,会给你镜像) +2. 头歌账号绑定邮箱,并且在本地准备Git +3. 在Obsidian里面安装插件Git +##### 详细步骤 +1. 下载Obsidian和Git For Windows(如果官网下不来,会给你镜像) +2. 打开头歌,打开个人主页,进入开发项目,新建一个项目 +![[edit1.png]] +新建项目的时候,如果没有绑定邮箱,会要求你绑定邮箱; +如果没有 +### 前置知识(需要在进入研讨组后尽快掌握) +1. KaTeX数学公式排版知识 +2. 基础Markdown知识 +3. 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(7分) $\begin{bmatrix}1&2^{101}-2&0\\0&2^{100}&0\\0&\frac{5}{3}(1-2^{100})&1\end{bmatrix}$ +>解析: +>$A$ 的特征值为 $1,1,2$, +>$A=\begin{bmatrix}0&1&2\\0&0&1\\1&-1&-\frac{5}{3}\end{bmatrix}\begin{bmatrix}1&0&0\\0&1&0\\0&0&2\end{bmatrix}\begin{bmatrix}0&1&2\\0&0&1\\1&-1&-\frac{5}{3}\end{bmatrix}^{-1}$ +2. (6分) $\frac{n(n-1)}{2}$ +>解析:略 +3. $s$ 是奇数 +>解析:向量组 $\beta_1,\beta_2,\cdots,\beta_s$ 线性无关,即 $\alpha_1,\alpha_2,\cdots,\alpha_s$ 到 $\beta_1,\beta_2,\cdots,\beta_s$ 的过渡矩阵可逆(此时两个向量组等价),过渡矩阵为$\begin{bmatrix}1&&&\cdots&1\\1&1&&&\\&1&1&&\\&&\ddots&\ddots&\\&&&1&1\end{bmatrix}_{s\times s}$,过渡矩阵的行列式不为$0$ +>按照第一行展开: +>$\begin{vmatrix}1&&&\cdots&1\\1&1&&&\\&1&1&&\\&&\ddots&\ddots&\\&&&1&1\end{vmatrix}_{s\times s}=(-1)^2\begin{vmatrix}1&&&&\\1&1&&&\\&1&1&&\\&&\ddots&\ddots&\\&&&1&1\end{vmatrix}_{(s-1)\times (s-1)}+(-1)^{s+1}\begin{vmatrix}1&1&&&\\&1&1&&\\&&1&\ddots&\\&&&\ddots&1\\&&&&1\end{vmatrix}_{(s-1)\times (s-1)}$ +>即$\begin{vmatrix}1&&&\cdots&1\\1&1&&&\\&1&1&&\\&&\ddots&\ddots&\\&&&1&1\end{vmatrix}_{s\times s}=1+(-1)^{s+1}=\begin{cases}0,s\text{为偶数}\\2,s\text{为奇数}\end{cases}$ +>所以 $s$ 是奇数 +4. (6分) 0 +>解析:因为$A,B$是正交矩阵且$|A|=-|B|$,所以$A^\mathrm{T}A=E, B^\mathrm{T}B=E, |A|=\pm 1, |B|=\mp 1$, +>而$A^{-1}(A+B)B^{-1}=B^{-1}+A^{-1}=A^\mathrm{T}+B^\mathrm{T}=(A+B)^\mathrm{T}$, +>所以$|A^{-1}||A+B||B^{-1}|=|(A+B)^\mathrm{T}|=|A+B|$,$|A^{-1}||B^{-1}|=-1$ +>所以$-|A+B|=|A+B|$,即$|A+B|=0$ +5. (7分) 0 +>$A\sim B\sim\begin{bmatrix}1&&&\\&2&&\\&&3&\\&&&4\end{bmatrix}$, +>所以$B^{-1}-E\sim\begin{bmatrix}1&&&\\&2&&\\&&3&\\&&&4\end{bmatrix}^{-1}-E=\begin{bmatrix}0&&&\\&-\frac{1}{2}&&\\&&-\frac{2}{3}&\\&&&-\frac{3}{4}\end{bmatrix}$, +>所以$|B^{-1}-E|=\begin{vmatrix}0&&&\\&-\frac{1}{2}&&\\&&-\frac{2}{3}&\\&&&-\frac{3}{4}\end{vmatrix}=0$ +6. (7分) 0 +>解析:设 $\alpha,\beta,\gamma$ 为 $x^3+px+q=0$ 的三个根,则 $x^3+px+q=$$(x-\alpha)(x-\beta)(x-\gamma)=x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\alpha\gamma+\beta\gamma)x+\alpha\beta\gamma$ ,所以$\alpha+\beta+\gamma=0$,将行列式第二、三行加在第一行,第一行全为$\alpha+\beta+\gamma=0$,故行列式为0 +7. (20分) $D_n=2^{n-1}(2+\sum\limits_{j=1}^n a_j^2)$ +>解析 +>方法1: +>行列式加边法:$D_n=\begin{vmatrix}1&a_1&a_2&\cdots&a_n\\0&2+a_1^2&a_1a_2&\cdots&a_1a_n\\0&a_2a_1&2+a_2^2&\cdots&a_2a_n\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&a_na_1&a_na_2&\cdots&2+a_n^2\end{vmatrix}$ +>将第一行乘以 $(-a_i)$ 加到第 $i+1$ 行: +>$D_n=\begin{vmatrix}1&a_1&a_2&\cdots&a_n\\-a_1&2&&&\\-a_2&&2&&\\\vdots&&&\ddots&\\-a_n&&&&2\end{vmatrix}$(模板:箭头行列式) +>再将第 $j+1$ 列乘以 $\frac{a_j}{2}$ 加到第一列: +>$D_n=\begin{vmatrix}1+\sum\limits_{j=1}^n\frac{a_j^2}{2}&a_1&a_2&\cdots&a_n\\&2&&&\\&&2&&\\&&&\ddots&\\&&&&2\end{vmatrix}$ +>所以$D_n=2^{n-1}(2+\sum\limits_{j=1}^n a_j^2)$ +>方法2: +>核心结论:$|E+AB|=|E+BA|$ +>$D_n=|2E_n+xx^\mathrm{T}|=2^n|E_n+\frac{1}{2}xx^\mathrm{T}|=2^n|E_1+\frac{1}{2}x^\mathrm{T}x|=2^n(1+x^\mathrm{T}x)=2^{n-1}(2+\sum\limits_{j=1}^{n}a_j^2)$ +8. (20分) $\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$ +>解析:由 $\alpha_3=2\alpha_1+\alpha_2$ 可得 $A$ 的列向量组线性相关, $|A|=0$;又因为 $A$ 的三个特征值各不相同,故 $A$ 有两个不为零的特征值 $\lambda_1,\lambda_2$,且 $A$ 可相似对角化,即 $A=P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P$,$\mathrm{rank}A=\mathrm{rank}(P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P)=\mathrm{rank}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}=2$ +>故 $Ax=0$ 的解空间维数是 $1$ (5分) +>$\beta=\alpha_1+3\alpha_2+4\alpha_3$,所以 $(1,3,4)^\mathrm{T}$ 为特解;(5分) +>$\alpha_3=2\alpha_1+\alpha_2$,所以$A\begin{bmatrix}2k\\k\\-k\end{bmatrix}=2\alpha_1+\alpha_2-\alpha_3=0$,所以 $(2,1,-1)^\mathrm{T}$ 为基础解系;(10分) +>解空间维数是 $1$ ,方程的解 $\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$ 维数是 $1$,该解完备 +9. 证明如下: +>(1)(10分) +> 对于方程组$Ax=0$ (a)和 $A^\mathrm{T}Ax=0$ (b),b的解空间一定包含a的解空间(5分); +>而方程b两边同时乘以$x^\mathrm{T}$,得 $x^\mathrm{T}A^\mathrm{T}Ax=0$ ,即 $(x^\mathrm{T}A^\mathrm{T})(Ax)=0 \to Ax=0$, +>所以a的解空间包含b的解空间(5分), +>所以a,b同解,所以$\mathrm{rank}A=\mathrm{rank}{A^\mathrm{T}A}$ +>(2) (10分) +>$A^\mathrm{T}Ax=A^\mathrm{T}\beta \iff \mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ +>而由(1)的结论得等式左边 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}A$; +>等式右边 $\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})\ge \mathrm{rank}A^\mathrm{T}A=\mathrm{rank}A$ (5分) +>又$\mathrm{rank}(A^\mathrm{T}\begin{bmatrix}A&\beta\end{bmatrix})\le \min{(\mathrm{rank}A^\mathrm{T},\ \mathrm{rank}\begin{bmatrix}A&\beta\end{bmatrix})}=\mathrm{rank}A$ (5分) +>所以$\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})=\mathrm{rank}A$ +>故 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ 得证. \ No newline at end of file diff --git a/编写小组/试卷/线代期末复习模拟/1.10线代限时练.md b/编写小组/试卷/线代期末复习模拟/1.10线代限时练.md index a837904..c76c6ad 100644 --- a/编写小组/试卷/线代期末复习模拟/1.10线代限时练.md +++ b/编写小组/试卷/线代期末复习模拟/1.10线代限时练.md @@ -1,13 +1,13 @@ 一、填空题 -1. 设 $A=\begin{bmatrix}1&2&0\\0&2&0\\-2&-1&-1\end{bmatrix}$,则$A^{100}=$\_\_\_\_\_\_\_\_\_\_\_ -2. 线性空间 $V = \{A\in\mathbb{R}^{n\times n}|A = -A^\mathrm{T}\}$ 的维数是\_\_\_\_\_\_\_\_\_\_\_ -3. 设向量组 $\alpha_1,\alpha_2,\cdots,\alpha_s$ 线性无关,$\beta_1=\alpha_1+\alpha_2,\beta_2=\alpha_2+\alpha_3,\cdots,\beta_{s-1}=\alpha_{s-1}+\alpha_s$,则向量组 $\beta_1,\beta_2,\cdots,\beta_s$ 线性无关的充要条件是\_\_\_\_\_\_\_\_\_\_\_ -4. 已知 $A,B$ 均为 $n$ 阶正交矩阵,且 $|A|=-|B|$,则 $|A+B|$ 的值为\_\_\_\_\_\_\_\_\_\_\_ -5. 已知 $4$ 阶矩阵 $A$ 与 $B$ 相似,$A$ 的全部特征值为 $1,2,3,4$,则行列式 $|B^{-1}-E|$ 为\_\_\_\_\_\_\_\_\_\_\_ -6. 设 $\alpha,\beta,\gamma$ 为 $x^3+px+q=0$ 的三个根,则行列式 $\begin{vmatrix}\alpha&\beta&\gamma\\\gamma&\alpha&\beta\\\beta&\gamma&\alpha\end{vmatrix}$ 的值为\_\_\_\_\_\_\_\_\_\_\_ +1. (7分)设 $A=\begin{bmatrix}1&2&0\\0&2&0\\-2&-1&-1\end{bmatrix}$,则$A^{100}=$\_\_\_\_\_\_\_\_\_\_\_ +2. (6分)线性空间 $V = \{A\in\mathbb{R}^{n\times n}|A = -A^\mathrm{T}\}$ 的维数是\_\_\_\_\_\_\_\_\_\_\_ +3. (7分)设向量组 $\alpha_1,\alpha_2,\cdots,\alpha_s$ 线性无关,$\beta_1=\alpha_1+\alpha_2,\beta_2=\alpha_2+\alpha_3,\cdots,\beta_{s-1}=\alpha_{s-1}+\alpha_s, \beta_{s}=\alpha_{s}+\alpha_1$,则向量组 $\beta_1,\beta_2,\cdots,\beta_s$ 线性无关的充要条件是\_\_\_\_\_\_\_\_\_\_\_ +4. (6分)知 $A,B$ 均为 $n$ 阶正交矩阵,且 $|A|=-|B|$,则 $|A+B|$ 的值为\_\_\_\_\_\_\_\_\_\_\_ +5. (7分)已知 $4$ 阶矩阵 $A$ 与 $B$ 相似,$A$ 的全部特征值为 $1,2,3,4$,则行列式 $|B^{-1}-E|$ 为\_\_\_\_\_\_\_\_\_\_\_ +6. (7分)设 $\alpha,\beta,\gamma$ 为 $x^3+px+q=0$ 的三个根,则行列式 $\begin{vmatrix}\alpha&\beta&\gamma\\\gamma&\alpha&\beta\\\beta&\gamma&\alpha\end{vmatrix}$ 的值为\_\_\_\_\_\_\_\_\_\_\_ 二、解答题 -7. 计算$n$阶行列式 $D_n=\begin{vmatrix}2+a_1^2 & a_1a_2 & \cdots&a_1a_n \\ a_2a_1& 2+a_2^2 & \cdots & a_2a_n \\ \vdots & \vdots & \ddots & \vdots \\ a_na_1&a_na_2&\cdots& 2+a_n^2\end{vmatrix}$. -8. 已知三阶方阵 $A=\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}$ 有三个不同的特征值,其中$\alpha_3=2\alpha_1+\alpha_2$,若 $\beta=\alpha_1+3\alpha_2+4\alpha_3$ ,求线性方程组 $Ax=\beta$ 的通解. -9. 设 $A$ 是 $m\times n$ 实矩阵, $\beta \neq 0$ 是 $m$ 维实列向量,证明: +7. (20分)计算$n$阶行列式 $D_n=\begin{vmatrix}2+a_1^2 & a_1a_2 & \cdots&a_1a_n \\ a_2a_1& 2+a_2^2 & \cdots & a_2a_n \\ \vdots & \vdots & \ddots & \vdots \\ a_na_1&a_na_2&\cdots& 2+a_n^2\end{vmatrix}$. +8. (20分)已知三阶方阵 $A=\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}$ 有三个不同的特征值,其中$\alpha_3=2\alpha_1+\alpha_2$,若 $\beta=\alpha_1+3\alpha_2+4\alpha_3$ ,求线性方程组 $Ax=\beta$ 的通解. +9. (20分)设 $A$ 是 $m\times n$ 实矩阵, $\beta \neq 0$ 是 $m$ 维实列向量,证明: (1) $\mathrm{rank}A=\mathrm{rank}(A^\mathrm{T}A)$ . (2) 线性方程组 $A^\mathrm{T}Ax = A^\mathrm{T}\beta$ 有解. \ No newline at end of file