vault backup: 2025-12-29 20:16:39

编写小组/试卷/2018-19线性代数期末考试卷.md

@@ -64,4 +64,60 @@
    (C) $a = 2, b = 0, c = 1$;  
    (D) $a = 2, b = 1, c = 2$.
 
+
+7. 已知向量 $\alpha_1 = (1,0,-1,0)^T$，$\alpha_2 = (1,1,-1,-1)^T$，$\alpha_3 = (-1,0,1,1)^T$，则向量 $\alpha_1 + 2\alpha_2$ 与 $2\alpha_1 + \alpha_3$ 的内积
+   $$
+   \langle \alpha_1 + 2\alpha_2,\, 2\alpha_1 + \alpha_3 \rangle = \underline{\qquad\qquad}.
+   $$
+
+8. 设二阶矩阵 $A$ 有两个相异特征值，$\alpha_1, \alpha_2$ 是 $A$ 的线性无关的特征向量，且 $A^2 (\alpha_1 + \alpha_2) = \alpha_1 + \alpha_2$，则
+   $$
+   |A| = \underline{\qquad\qquad}.
+   $$
+
+9. 若向量组
+   $$
+   \alpha_1 = (1,0,1)^T,\quad \alpha_2 = (0,1,1)^T,\quad \alpha_3 = (1,3,5)^T
+   $$
+   不能由向量组
+   $$
+   \beta_1 = (1,1,1)^T,\quad \beta_2 = (1,2,3)^T,\quad \beta_3 = (3,4,a)^T
+   $$
+   线性表示，则
+   $$
+   a = \underline{\qquad\qquad}.
+   $$
+
+10. 设矩阵
+    $$
+    A = \begin{bmatrix}
+    1 & a_1 & a_1^2 & a_1^3 \\
+    1 & a_2 & a_2^2 & a_2^3 \\
+    1 & a_3 & a_3^2 & a_3^3 \\
+    1 & a_4 & a_4^2 & a_4^3
+    \end{bmatrix},\quad
+    x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix},\quad
+    b = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix},
+    $$
+    其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为
+    $$
+    \underline{\qquad\qquad\qquad\qquad}.
+    $$
+
+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
+    $$
+    \underline{\qquad\qquad\qquad\qquad}.
+    $$
+
 ---