2024秋线性代数期中-回忆版

1. Multiple Choice.Only one choice is correct

1-1

Suppose $A$ is an $m\times n$ real matrix with $m<n$, and the equation $Ax=b$ has a solution for any $m$-dimensional real column vector $b$. Which of the following assertions must be true?

(A) $Ax=b$ has a unique solution for every $b\in R^m$

(B) $A^{\mathrm{\top }}x=d$ has a solution for any $d\in R^n$

(C) $N(A)=\{0\}$

(D) A has a right inverse

1-2

Suppose we have matrices $A,B$ satisfying $EA=B$ for some invertible matrix $E$. Which of the following assertions must be true?

(A) $N(A)=N(B)$

(B) $C(A)=C(B)$

(C) $N\left(A^{\mathrm{\top }}\right)=N\left(B^{\mathrm{\top }}\right)$

(D) $A^{\mathrm{\top }}A=B^{\mathrm{\top }}B$

1-3

For any $m\times n$ matrix $A$ with reduced row echelon from $U$. Which of the following assertions must be true?

(A) $C(A)=C\left(A^{\mathrm{\top }}A\right)$

(B) $\mathrm{rank}(A)=\mathrm{rank}\left(A^{\mathrm{\top }}A\right)$

(C) $C(A)=C(U)$

(D) If $C(A)=R^m$, then $A^{\mathrm{\top }}A$ is invertible

1-4

Let $A,B,C$ be $n\times n$ matrices such that $ABC=I$, where $I$ is the identity matrix of order $n$, then

(A) $ACB=I$

(B) $CBA=I$

(C) $BCA=I$

(D) $BAC=I$

1-5

Let $A$ be an $n\times n$ matrix $(n>1)$ such that $A=A^2$, and $I$ be the $n\times n$ identity matrix, then

(A) $\mathrm{rank}(A)+\mathrm{rank}(A-I)>n$

(B) $\mathrm{rank}(A)+\mathrm{rank}(A-I)<n$

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

(D) $\mathrm{rank}(A)+\mathrm{rank}(A-I)=n-1$

2. Fill in the blanks

2-1

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

2-2

Let $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& -1& 1\\ 3& 1& 3\end{array}\right]$. The LDU factorization of $A$ has $L=$ $\underset{―}{}$

2-3

The matrix which projects every vector $b$ in $R^3$ onto the line in the direction of $\left[\begin{array}{l}2\\ 3\\ 1\end{array}\right]$ through the origin is $\underset{―}{}$

2-4

Let $A$ be a $2024\times 2025$ real matrix with $\dim N(A)=11$, then $\dim N\left(A^{\mathrm{\top }}\right)=$ $\underset{―}{}$

3

Let

$$A=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& -1\\ 0& 1& 0\end{array}\right]$$

Consider $X-X{A}^2-AX+AX{A}^2=I$,

where $I$ is the $3\times 3$ identity matrix, and $X$ is a $3\times 3$ matrix

3-a

Compute $I-A$ and $I-A^2$

3-b

Find all possible $X$

4

Let

$$A=\left[\begin{array}{llll}1& 0& 2& 0\\ 1& 1& 3& 0\\ 1& 0& 2& 1\\ 3& 2& 8& 0\end{array}\right],b=\left[\begin{array}{l}3\\ 3\\ 3\\ 9\end{array}\right]$$

4-a

Find the reduced row echelon form of $A$

4-b

Find a basis for the row space $C\left(A^{\mathrm{\top }}\right)$, the column space $C(A)$, and the left nullspace $N\left(A^{\mathrm{\top }}\right)$

4-c

Find the complete solution to $Ax=b$. In other words, find all the solutions to $Ax=b$

5

Consider the following subspace of $R^3$:

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

5-a

Show that:

$$V_1=\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right],V_2=\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$$

is a basis of $V$

5-b

Let $T$ be the linear transformation from $V$ to $R^3$ defined as follows:

$$T\left(\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]\right)=\left[\begin{array}{c}2x_1-x_2\\ x_2+x_3\\ x_1\end{array}\right]$$

Find the matrix representation of $T$ with respect to the ordered basis $v_1,v_2$ of $V$ and the ordered basis

$$e_1=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right],e_2=\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right],e_3=\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right]$$

5-c

Can we find a vector $v\in V$ such that $T(v)=\left[\begin{array}{l}3\\ 2\\ 1\end{array}\right]$ of $R^3$? If so, find one such $v$.

Otherwise give an explanation.

6

Let $A=\left[\begin{array}{rrr}1& 1& 1\\ 0& 0& 1\\ 0& 0& -1\end{array}\right]$ and $U=\{B\in M_{3\times 3}(R)\mid AB-BA\}$,

where $M_{3\times 3}(R)$ denotes the vector space of all $3\times 3$ real matrices with the ordinary matrix addition and scalar multiplication

6-a

Show that $U$ is a subspace of $M_{3\times 3}(R)$

6-b

Find a basis of $U$

6-c

Find the dimension of $U$

7

Prove the following two independent statements

7-a

Let $A$ be an $m\times n$ real matrix. Suppose $v_1,v_2,\cdots ,v_s$ is a basis of $C\left(A^{\mathrm{\top }}\right)$ and $w_1,w_2,\cdots ,w_t$ is a basis of $N(A)$. Show that: $s+t=n$ and

$$v_1,v_2,\dots ,v_s,w_1,w_2,\cdots ,w_t$$

is a basis of $R^n$

7-b

Let $A$ be an $m\times n$ real matrix with rank $m$. Show that $A{A}^{\mathrm{\top }}$ is invertible