2024-2025学年春季学期线性代数期中考试试卷

考试科目: 线性代数

考试时长: 120 分钟

开课单位: 数 学 系

命题教师: 线性代数教学团队

本试卷共(7) 大题, 满分(100) 分 (考试结束后请将试卷, 答题本, 草稿纸一起交给监考老师)

This exam paper contains 7$ questions and the score is 100 in total (Please hand in your exam paper, answer sheet, and your scrap paper to the proctor when the exam ends)

$C(A)$ : the column space of matrix A

$C(A^T)$ : the row space of matrix A

$N(A)$ : the nullspace of matrix A

$N(A^T)$ : the left nullspace of matrix A

O : the zero matrix

rank A : the rank of matrix A

一 (15 points, 3 points each) Multiple Choice.Only one choice is correct

(共 15 分, 每小题 3 分) 选择题, 只有一个选项是正确的

1-1

Let $A$ be a $3\times 4$ matrix with rank $A=2$ and

$$P=\left[\begin{array}{lll}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\\ 1& 2& 3\end{array}\right]$$

then $\mathrm{rank}PA=$

设 $A$ 为一个秩为 2 的 $3\times 4$ 矩阵,

$$P=\left[\begin{array}{lll}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\\ 1& 2& 3\end{array}\right]$$

则 $\mathrm{rank}PA=$

(A) 1

(P) 2

(C) 3

(D) 4

1-2

Which of the following subsets of ${\mathbb{R}}^3$ is a subspace of ${\mathbb{R}}^3$ ?

下列集合构成 ${\mathbb{R}}^3$ 的子空间的是?

(A) $V_1=\{\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]\in {\mathbb{R}}^3|,x_1+x_2+x_3=1\}$

(B) $V_2=\{\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]\in {\mathbb{R}}^3|,x_1^2+x_2^2-x_3^2=0\}$

(C) $V_3=\{\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]\in {\mathbb{R}}^3|,x_1+x_2+x_3=0\}$

(B) $V_4=\{\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]\in {\mathbb{R}}^3|,x_1\ge 0,x_2\ge 0\}$

1-3

Let $u,v\in {\mathbb{R}}^4,\lambda \in \mathbb{R}$. which of the following assertions is false?

设 $u,v\in {\mathbb{R}}^4,\lambda \in \mathbb{R}$. 以下哪个说法是错误的?

(A) Suppose $u,v$ are nonzero vectors and $u^{T}v=0$. then $u,v$ are linearly independent

(B) $\parallel u+v{\parallel }^2+\parallel u-v{\parallel }^2=2(\parallel u{\parallel }^2+\parallel v{\parallel }^2)$

(C) $\lambda u=0$. then $\lambda =0$ or $u=0$

(D) $u^{T}v=0$. then $u=0$ or $v=0$

1-4

$${\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]}^{2024}\left[\begin{array}{lll}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]{\left[\begin{array}{lll}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]}^{2025}=$$

(A) $\left[\begin{array}{ccc}2& 1& 3\\ 5& 4& 6\\ 4056& 2031& 6081\end{array}\right]$

(B) $\left[\begin{array}{ccc}5& 4& 6\\ 2& 1& 3\\ 4056& 2031& 6081\end{array}\right]$

(D) $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 2031& 4056& 6081\end{array}\right]$

(D) $\left[\begin{array}{ccc}12148& 5& 6\\ 6073& 2& 3\\ 18223& 8& 9\end{array}\right]$

1-5

Let $A$ be an $m\times n$ real matrix and $b\in {\mathbb{R}}^m$. Suppose $Ax=b$ has infinitely many solutions, which of the following assertions must be true?

设 $A$ 为 $m\times n$ 实矩阵, $b\in {\mathbb{R}}^m$如果 $Ax=b$ 有无穷多解, 下列哪个结论一定是正确的?

(A) $A$ is a square matrix

(B) $\mathrm{rank}A=m$

(C) $\mathrm{rank}A=n$

(D) $\mathrm{rank}A<n$

二 (20 points, 5 points each) Fill in the blanks

(共 20 分, 每小题 5 分) 填空题

2-1

Let $A=\left[\begin{array}{lll}1& 6& 4\\ 0& 3& 2\\ 0& 0& 2\end{array}\right]$Then $A^{-1}=\underset{―}{}$

2-2

Let $A=\left[\begin{array}{rrr}1& 2& -2\\ 4& t& 3\\ 3& -1& 1\end{array}\right]$ and $B$ be a $3\times 3$ nonzero matrix such that $AB=O$Then $t=\underset{―}{}$

2-3

Let $A=\left[\begin{array}{rrr}1& 2& 3\\ 2& 4& 6\\ 3& 8& 7\\ 5& 12& 13\end{array}\right]$A basis of $N(A)$ is $\underset{―}{}$

2-4

Let $A=\left[\begin{array}{llll}1& 1& 1& 1\\ 2& 2& 2& 4\\ 1& 1& 2& 3\\ 3& 3& 5& 8\end{array}\right]$ Then $\dim C(A)=\underset{―}{}$

三 (10 points)

Find an $LU$ factorization of the matrix

求下列矩阵的一个 $LU$ 分解:

$$\left[\begin{array}{ccc}4& -5& 6\\ 8& -6& 7\\ 12& -7& 12\end{array}\right]$$

四 (20 points)

Consider the following system of linear equations

$$(I):\{\begin{array}{r}{x}_1+x_2+x_3+x_4+x_5=1\\ 3x_1+2x_2+x_3+x_4-3x_5=\lambda \\ x_2+2x_3+2x_4+6x_5=3\\ 5x_1+4x_2+3x_3+3x_4-x_5=\mu \end{array}$$

(A) For what values of $\lambda$ and $\mu$ does the system( $I$ ) have no solution or infinitely many solutions.当 $\lambda ,\mu$ 满足什么条件时, 上述线性方程组( $I$ ) 无解, 有无穷多解?

(B) Solve for all the solutions of $(I)$ if the system is consistent.在方程组(I) 有解时, 求出其通解

五 (20 points)

Let $V={\mathbb{R}}^{2\times 2}$ be the vector space of all $2\times 2$ real matrices.Define a map as follows:

$$T:V\to V,T(A)=A+A^T,\mathrm{\forall }A\in V$$

(A) Show that $T$ is a linear transformation.证明: $T$ 为线性变换

(B) Let kernel $T=\{A\in V\mid T(A)=O\}$. where $O$ denotes the $2\times 2$ zero matrix.Show that kernel $T$ is a subspace of $V$ and find a basis for kernel $T$

设 kernel $T=\{A\in V\mid T(A)=O\}$. 这里 $O$ 表示 2 阶零矩阵.证明 kernel $T$ 为 $V$ 的一个子空间, 并求 kernel $T$ 的一个基向量组

(C) Find the matrix representation of $T$ with respect to the following ordered basis of $V$ :

求线性变换 $T$ 在如下 $V$ 的有序基下的矩阵表示:

$$\left[\begin{array}{ll}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{ll}0& 1\\ 0& 0\end{array}\right],\left[\begin{array}{ll}0& 0\\ 1& 0\end{array}\right],\left[\begin{array}{ll}0& 0\\ 0& 1\end{array}\right]$$

(D) Find all matrices $A$ such that $T(A)=\left[\begin{array}{ll}1& 2\\ 2& 1\end{array}\right]$

求所有满足 $T(A)=\left[\begin{array}{ll}1& 2\\ 2& 1\end{array}\right]$ 的矩阵 $A$

六 (10 points)

In physics, Hooke's law states that(within certain limits) there is a linear relationship between the length $x$ of a spring and the force $y$ applied to(or exerted by) the spring. That is, $y=cx+d$. where $c$ is called the spring constant.Use the following data to estimate the spring constant(the length is given in inches and the force is give in pounds)

在物理学中, Hooke's law 说的是(在有限范围内) 作用在弹簧上的力 $y$ 和弹簧的长度 $x$ 存在线性关系.也就是说, $y=cx+d$. 其中 $c$ 称之为弹簧常数.请用下列表格提供的数据估计弹簧常数(长度的单位是英寸, 力的单位是磅)

length	force
x	y
3.5	1.0
4.0	2.2
4.5	2.8
2.0	4.3

七 (5 points)

Let $A,B$ be $5\times 6$ real matrices with rank $A=2$. $\mathrm{rank}B=3$. and $\mathrm{rank}(A+B)=5$ Show that there exist $5\times 5$ invertible matrix $P$ and $6\times 6$ invertible matrix $Q$ such that

设 $A,B$ 为 $5\times 6$ 实矩阵, 且 $\mathrm{rank}A=2,\mathrm{rank}B=3$. 以及 $\mathrm{rank}(A+B)=5$

证明: 存在 5 阶可逆矩阵 $P$ 和 6 阶可逆矩阵 $Q$ 使得

$$PAQ=\left[\begin{array}{llllll}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\text{ and }PBQ=\left[\begin{array}{llllll}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$