2020 线性代数期中(回忆版)

1.Multiple Choice.Only one choice is correct

(1)

Let $A$ be $m\times n$ matrix and $b$ be a column vector in $R^m$ .Which of the following statements is correct?

(A) If $Ax=b$ has infinite many solutions, then $Ax=0$ has a nonzero solution

(B) If the system $Ax=0$ has only zero solution, then $Ax=b$ has one and only one solution

(C) If the rank of $A$ is $n$, then the system $Ax=b$ must have a solution

(D) If $A$ is a square matrix(i.e, $m=n$ ), then the system $Ax=b$ is consistent if and only if $A$ is invertible

(2)

Suppose $A$ is an $m\times n$ matrix, $B$ is an $n\times m$ matrix, and $I$ is the $m\times m$ identity matrix.If $AB=I$, then( )

$(A)$ the column vectors of $A$ are linearly independent, and the row vectors of $B$ are linearly independent

$(B)$ the column vectors of $A$ are linearly independent, and the column vectors of $B$ are linearly independent

(C) the row vectors of $A$ are linearly independent, and the column vectors of $B$ are linearly independent

(D) the row vectors of $A$ are linearly independent, and the row vectors of $B$ are linearly independent

(3)

Let $A$ be a $3\times 3$ matrix, and let $B$ be the matrix formed by adding the second column of $A$ to its first column.Suppose that after exchanging the second and third rows of $B$, the resulting matrix is the $3\times 3$ identity matrix.Let $P_1=\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right],P_2=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$

Then $A=()$

(A) $P_1{P}_2$

(B) $P_1^{-1}{P}_2$

(C) $P_2{P}_1$

(D) $P_2{P}_1^{-1}$

(4)

Let $A=\left[\begin{array}{ccc}-1& 2& 3\\ -3& 6& 8\\ 2& -4& t\end{array}\right]$, where $t\in R$ .Suppose $\mathrm{rank}(A)=2$ .Then( )

(A) $t=-6$

(B) $t=6$

(C) $t\ne 0$

(D) $t$ can be any real number

(5)

Which of the following statement is a incorrect? ( )

(A) For any matrix $A$, $\mathrm{rank}(A)=\dim C(A)$

(B) If $v_1,\cdots ,v_m$ are pairwise orthogonal nonzero vectors, then the vectors $v_1,\cdots ,v_m$ are linear independent

(C) If $A$ is an upper triangular $n\times n$ matrix such that $A^2=0$, then $A=0$

(D) Let $A,B$ be $n\times n$ matrices such that $AB$ is invertible.Then both $A$ and $B$ are invertible

2. Fill in the blanks

(I) Let $A,B$ be invertible $n\times n$ matrices. Then the inverse of the block matrix $\left[\begin{array}{ll}0& A\\ B& 0\end{array}\right]$ is $\underset{―}{}$

(2) Suppose $A$ is a $3\times 4$ matrix and $\dim N(A)=2$. Then $\dim N\left(A^{\mathrm{\top }}\right)=$ $\underset{―}{}$

(3) Let $A=\left[\begin{array}{lll}1& 0& 0\\ a& 1& 0\\ b& c& 1\end{array}\right]$. Then $A^{-1}=$ $\underset{―}{}$

(4) Let $u,v$ be vectors in $R^n$ such that $\parallel u\parallel =3,\parallel v\parallel =4$ and $u^{\mathrm{\top }}v=-3$

Then $\parallel 2u+3v\parallel =$

(5) Let $A=\left[\begin{array}{cc}1& 1\\ 1& 0\\ 0& -1\end{array}\right],b=\left[\begin{array}{l}1\\ 2\\ 7\end{array}\right]$. Then the least squares solution to $Ax=b$ is $\hat{x}=$ $\underset{―}{}$

3

Find the $LU$ factorization of the matrix $A=\left[\begin{array}{lll}3& 1& 1\\ 1& 3& 1\\ 1& 1& 3\end{array}\right]$

4

Let $A=\left[\begin{array}{lllll}0& 1& 2& 3& 4\\ 0& 1& 2& 4& 6\\ 0& 0& 0& 1& 2\end{array}\right]$. Please give a basis ofor each of the four fundamental subspaces

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

5

Let $E=\{u_1,u_2,u_3\}$ and $F=\{v_1,v_2\}$, where

$$u_1=\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right],u_2=\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right],u_3=\left[\begin{array}{c}-1\\ 1\\ 1\end{array}\right]\text{ and }{v}_1=\left[\begin{array}{c}1\\ -1\end{array}\right],v_2=\left[\begin{array}{c}2\\ -1\end{array}\right]\text{. }$$

Define the linear transformation $T:R^3\to R^2$ by

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

Find the matrix $A$ representing $T$ with respect to the ordered basses $E$ and $F$

6

Let $A,B$ be $n\times n$ matrices. Suppose $A$ and $B$ are both symmetric. Is $AB$ necessarily symmetric

If yes, please give a proof. Otherwise please give a counter example

7

The following two questions are independent:

(A) Let $A$ be the $2\times 2$ matrix such that the linear transformation $R^2\to R^2,\nu \mapsto A\nu$ rotates every vector in $R^2$ through ${60}^{\circ }$ counter-dockwise (about the origin)

Find $A$ and $A^{2020}$

(B) Three planes ${\mathbb{I}}_1,{\mathbb{I}}_2,{\mathbb{I}}_3$ in the space $R^3$ are given by the equations

$$\begin{array}{rl}& {\mathbb{I}}_1:x+y+z=0,\\ & {\mathbb{I}}_2:2x-y+4z=0,\\ & {\mathbb{I}}_3:-x+2y-z=0\end{array}$$

Determine a matrix representative (in the standard basis of $R^3$ ) of a linear transformation taking the $xy$ plane to ${\mathbb{I}}_1$, the ez plane to ${\mathbb{I}}_2$ and the $zx$ plane to ${\mathbb{I}}_3$

8

Let $A$ be a $3\times 3$ matrix such that $\mathrm{rank}(A)=2$ and $A^3=0$

(A) Prove that rank $\left(A^2\right)=1$

(B) Let ${\alpha }_1\in R^3$ be a nonzero vector such that $A{\alpha }_1=0$. Prove that there exist vectors ${\alpha }_2,{\alpha }_3$ such that $A{\alpha }_2={\alpha }_1,A^2{\alpha }_3={\alpha }_1$

(C) For any vectors ${\alpha }_2,{\alpha }_3$ described as above, show that ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are linearly independent

(In this problem, you are allowed to assume the statements of some questions to answer subsequent questions.)