From 15c2439b3e604098544902e8e20b947ebbcf892d Mon Sep 17 00:00:00 2001
From: Elwood <3286545699@qq.com>
Date: Mon, 29 Dec 2025 21:22:08 +0800
Subject: [PATCH] vault backup: 2025-12-29 21:22:08

---
 .../试卷/1231线性代数考试卷.md       | 46 ++++++++++++++-----
 1 file changed, 35 insertions(+), 11 deletions(-)

diff --git a/编写小组/试卷/1231线性代数考试卷.md b/编写小组/试卷/1231线性代数考试卷.md
index d618d20..e2eb5ce 100644
--- a/编写小组/试卷/1231线性代数考试卷.md
+++ b/编写小组/试卷/1231线性代数考试卷.md
@@ -31,6 +31,38 @@ $$
 -5 & 1
 \end{bmatrix}.
 $$
+
+---
+
+解析：
+
+$$
+\varepsilon_1 = e_1 + 5e_2,\quad \varepsilon_2 = 0e_1 + 1e_2.
+$$
+
+把它们按列排成矩阵形式：
+
+$$
+[\varepsilon_1, \varepsilon_2] = [e_1, e_2] 
+\begin{pmatrix}
+1 & 0 \\
+5 & 1
+\end{pmatrix}.
+$$
+
+基变换矩阵为：
+
+$$
+T = \begin{pmatrix} 1 & 0 \\ 5 & 1 \end{pmatrix}.
+$$
+$$
+\quad 
+T^{-1} = \begin{pmatrix} 1 & 0 \\ -5 & 1 \end{pmatrix}.
+$$
+$\quad T^{-1} = \begin{pmatrix} 1 & 0 \\ -5 & 1 \end{pmatrix}$是坐标变换矩阵，即为过渡矩阵，选D
+
+---
+
 3. 设向量组  
    $$
    \alpha_1 = (0, 0, c_1)^T,\quad 
@@ -143,18 +175,10 @@ $$A^n = 6^{n-1}\begin{bmatrix}3&-1\\-9&3\end{bmatrix} $$
     \underline{\qquad\qquad\qquad\qquad}.
     $$
 
-11. 若 $n$ 阶实对称矩阵 $A$ 的特征值为
-    $$
-    \lambda_i = (-1)^i \quad (i=1,2,\dots,n),
-    $$
-    则
-    $$
-    A^{100} = \underline{\qquad\qquad\qquad\qquad}.
-    $$
 
-12. 设 $n$ 阶矩阵 $A = [a_{ij}]_{n \times n}$，则二次型
-    $f(x_1, x_2, \dots, x_n) = \sum_{i=1}^n (a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n)^2$
-    的矩阵为
+
+11. 
+12.
     $$
     \underline{\qquad\qquad\qquad\qquad}.
     $$