22-23秋线性代数期中(回忆版)

1.Multiple Choice.Only one choice is correct

(1)

Which of the following statements can guarantee that the following homogeneous system of linear equations $\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=0\\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n=0\end{array}$ has a nonzero solution?

(A) $m\le n$

(B) $m=n$

(C) $m>n$

(D) The rank of the coefficient matrix is less than $n$

(2)

Which of the following matrices can be written as a product of elementary matrices?

(A) $\left[\begin{array}{lll}1& 2& 3\\ 0& 4& 2\end{array}\right]$

(B) $\left[\begin{array}{ccc}1& 2& 0\\ 0& -1& 3\\ 0& 0& 2\end{array}\right]$

(C) $\left[\begin{array}{lll}1& 0& 1\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$

(D) $\left[\begin{array}{ccc}-3& 2& 7\\ -1& 2& 3\\ 0& -2& -1\end{array}\right]$

(3)

Let ${\beta }_1,{\beta }_2,{\beta }_3$ be a basis of the null space $N(A)$ of some matrix $A$ .Another basis of $N(A)$ is ()

(A) ${\beta }_1+{\beta }_2,{\beta }_2+{\beta }_3,{\beta }_3+{\beta }_1$

(B) ${\beta }_1+{\beta }_2,{\beta }_2+{\beta }_3,{\beta }_3-{\beta }_1$

(C) ${\beta }_1-{\beta }_2,{\beta }_2-{\beta }_3,{\beta }_3-{\beta }_1$

(D) ${\beta }_1+2{\beta }_2,2{\beta }_2+3{\beta }_3,3{\beta }_3-{\beta }_1$

(4)

Let $a,b\in R$ .The set

$$V=\{(x,y,zw)=x+2y+3z+4w=a+b+1,x-2y+4z-w=a-2b-5\}$$

is a subspace of $R^4$ if

(A) $a=-1,b=1$

(B) $a=-2,b=1$

(C) $a=1,b=-2$

(D) $a=1,b=-1$

(5)

Let $u$ and $v$ be unit vectors in $R^3$ .If the vectors $u+2v$ and $5u-4v$ are orthogonal, then the angle $\alpha$ between $u$ and $v$ is( )

(A) $\alpha =\frac{\pi }{6}$

(B) $\alpha =\frac{\pi }{4}$

(C) $\alpha =\frac{\pi }{3}$

(D) $\alpha =\frac{3\pi }{4}$

2.Fill in the blanks

2-1

Let $A$ be a $5\times 8$ real matrix.If $\dim N(A)=3$, then $\dim \left(N\left(A^{\mathrm{\top }}\right)\right)=$ $\underset{―}{}$

2-2

All the $2\times 2$ matrices that commute with $\left[\begin{array}{ll}1& 0\\ 1& 1\end{array}\right]$ can be written in the form $\underset{―}{}$

2-3

Let $A=\left[\begin{array}{ccc}1& 2& -2\\ 2& -1& \lambda \\ 3& 1& -1\end{array}\right]$ .If $AB=0$ for some nonzero matrix $B$, then $\lambda =$ $\underset{―}{}$

2-4

An $LU$-factorization of $A=\left[\begin{array}{ll}2& 1\\ 3& 7\end{array}\right]$ is $L=$ $\underset{―}{}$ , $U=\underset{―}{}$

2-5

Suppose $b$ is a nonzero column vector.If ${\eta }_1,{\eta }_2$ are solutions to the system of linear equations $Ax=b$, and ${\lambda }_1{\eta }_1+{\lambda }_2{\eta }_2$ is another solution to $Ax=b$, then ${\lambda }_1,{\lambda }_2$ must satisfy $\underset{―}{}$

3

An audio processing company develops technology for mobile devies and is proud of the capacity of their products to filter surrounding noise.Here is a simplified model(a single-layer neural network) showing how it works. Let $s_1,s_2$ be the volumes of a pair of speakers and $n$ denote that of noise. Use 3 microphones to receive single signals with recorded volumes $m_1,m_2$,and $m_3$. All values are in decribels and shown in the following diagram, where the linear factor hiv indicates the rate of decay along each channel. (When a sound of 100 dB is transmitted along a channel with rate of decay $h$, the volume received is $100\mathrm{h}\mathrm{dB})$

graph TD
    S1[Speakers S1: 20 dB] -->|h11| M1[Microphone M1: 35 dB]
    S1 -->|h12| M2[Microphone M2: 20 dB]
    S1 -->|h13| M3[Microphone M3: 25 dB]

    S2[Speakers S2: Unknown dB] -->|h21| M1
    S2 -->|h22| M2
    S2 -->|h23| M3

    N[Noise n] -->|h31| M1
    N -->|h32| M2
    N -->|h33| M3

Suppose we are given the matrix $\left[h_{ij}\right]=\left[\begin{array}{ccc}0.875& 0.5& 0.75\\ 0.25& 0.5& 0.5\\ 0.625& 0.375& 0.5\end{array}\right]$. Estimate the volume of the unknown sourse speaker by solving a linear system for $s_2$

4

Let $T$ be a linear transformation from $R^3$ to $R^3$ such that

$$T\left({\alpha }_1\right)=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right],T\left({\alpha }_2\right)=\left[\begin{array}{l}1\\ 1\\ 2\end{array}\right],T\left({\alpha }_3\right)=\left[\begin{array}{l}0\\ 1\\ 1\end{array}\right]\text{ where }{\alpha }_1=\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right],{\alpha }_2=\left[\begin{array}{l}2\\ 1\\ 3\end{array}\right],{\alpha }_3=\left[\begin{array}{l}0\\ 2\\ 1\end{array}\right]\text{. }$$

4-(a)

Show that ${\alpha }_1,{\alpha }_2,{\alpha }_3$ is a basis of $R^3$

4-(b)

Find the representation matrix $A$ of $T$ (in the standard basis $e_1,e_2,e_3$ of $R^3$ )

4-(c)

Is the matrix $A$ invertible? Why?

5

Let $L$ be the line of intersection of $x_1+x_2+x_3=0$ and $2x_1-x_2-2x_3=0$ in $R^3$

Find the orthogonal projection of $b=\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]$ onto $L$

6

Let $u,v$ be nonzero column vectors in $R^n$ and $A=u{v}^{\mathrm{\top }}$

(A) Prove that the rank of $A$ is 1

(B) What are the possible values of the rank of the matrix $\left[\begin{array}{cc}{u}^{\mathrm{\top }}v& 0\\ 0& \nu u^{\mathrm{\top }}\end{array}\right]$ ? Justify your answer

7

Let ${\alpha }_1,\cdots ,{\alpha }_n$ be column vectors in $R^3$. Suppose that the systerm ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{n-1}$ is linearly dependent, and that the system ${\alpha }_2,{\alpha }_3,\cdots ,{\alpha }_n$ is linearly independent

Let $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\cdots ,{\alpha }_n)$ and $\beta ={\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n$

(A) Show that ${\alpha }_1$ can be written as a linear system combination of ${\alpha }_2,{\alpha }_3,\cdots ,{\alpha }_n,i.e$, there exist contents $k_2,k_3,\cdots ,k_n$ so that ${\alpha }_1=k_2{\alpha }_2+k_3{\alpha }_3+\cdots +k_n{\alpha }_n$

(B) Show that the linear system $Ax=\beta$ has infinitely many solutions

(C) Prove that if $n>2$, then $A^2\ne 0$. Here 0 denotes the zero matrix of order $n$