2023秋线性代数期中(回忆版)

2023.11

一 Multiple Choice.Only one choice is correct

1-1

Suppose that ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are a basis for the nullspace of a matrix $A,N(A)$ .Which of the following lists of vectors is also a basis of or $N(A)$ ?

(A) ${\alpha }_1+{\alpha }_2-{\alpha }_3,{\alpha }_1+{\alpha }_2+5{\alpha }_3,4{\alpha }_1+{\alpha }_2-2{\alpha }_3$

(B) ${\alpha }_1+2{\alpha }_2+{\alpha }_3,2{\alpha }_1+{\alpha }_2+2{\alpha }_3,{\alpha }_3+{\alpha }_1+{\alpha }_2$

(C) ${\alpha }_1+{\alpha }_2,{\alpha }_1+{\alpha }_2+{\alpha }_3$

(D) ${\alpha }_1-{\alpha }_2,{\alpha }_2-{\alpha }_3,{\alpha }_3-{\alpha }_1$

1-(2) Which of the following statements is correct?

(A) If the columns of $A$ are linearly independent, then $Ax=b$ has exactly one solution for every $b$

(B) Any $5\times 7$ matrix has linearly dependent columns

(C) If the columns of a matrix $A$ are linearly dependent, so are the rows

(D) The column space and row space of a $10\times 12$ matrix may have differed dimensions

1-3

Let

$${\alpha }_1=\left[\begin{array}{l}1\\ 4\\ 1\end{array}\right],{\alpha }_2=\left[\begin{array}{c}2\\ 1\\ -5\end{array}\right],{\alpha }_3=\left[\begin{array}{c}6\\ 2\\ -16\end{array}\right],\beta =\left[\begin{array}{c}2\\ t\\ 3\end{array}\right]$$

$\beta$ can be writtan as a linear combination of ${\alpha }_1,{\alpha }_2,{\alpha }_3$ if $t=($ )

(A) 1

(B) 3

(C) 6

(D) 9

1-(4) Which of the following statements is correct?

(A) Suppose that $EA=B$ and $E$ is an invertible matrix, then the column space of $A$ and the column space of $B$ are the same

(B) Let $A$ be $n\times n$ matrix with rank 1, then $A^n=cA$, where $n$ is a positive integer and $c$ is a real number

(C) Let $A,B$ be symmetric matrices, then $AB$ is symmetric

(D) If $A$ is of full row rank, then $Ax=0$ has only the zero solution

1-5

Let $A$ and $B$ be two $n\times n$ matrices.If $A$ is a non-zero matrix and $AB=0$, then

(A) $BA=0$

(B) $B=0$

(C) $(A+B)(A-B)=A^2-B^2$

(D) rank $B<n$

二 Fill in the blanks

2-1

Denote the vector space of $7\times 7$ real matrices by $M_{7\times 7}(R)$, and let $W$ be the subspace of $M_{7\times 7}(R)$ consisting of skew-symmetric real matrices, then $\dim W=$ $\underset{―}{}$. $A$ matrix $A$ is called skew symmetric if $A^{\mathrm{\top }}=-A$

2-2

Let $A,B$ be two $n\times n$ invertible matrices.Suppose the inverse of $\left[\begin{array}{ll}A& C\\ 0& B\end{array}\right]$ is $\left[\begin{array}{ll}{A}^{-1}& D\\ 0& B^{-1}\end{array}\right]$, where $O$ is the $n\times n$ zero matrix.Then $D=\underset{―}{}$

2-3

Let $A=\left[\begin{array}{llll}a& 1& 1& 1\\ 1& a& 1& 1\\ 1& 1& a& 1\\ 1& 1& 1& a\end{array}\right]$ with $\mathrm{rank}(A)<4$. Then $a=\underset{―}{}$

2-4

Consider the system of linear equations:

$$Ax=b=\{\begin{array}{l}x+2y=1\\ x-y=2\\ y=-1\end{array}$$

The least-squares solution for the system is $\underset{―}{}$

2-5

Let $H$ be the subspace of $R^3$ be defined as follows:

$$H=\{\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right],x_1+2x_2+x_3=0\}$$

$A$ unit vector orthogonal to $H$ is $\underset{―}{}$

三

Consider the following $4\times 5$ matrix $A$ with its reduced row echelon form $R$ :

$$A=\left[\begin{array}{ccccc}1& 2& \ast & 1& \ast \\ 0& 1& \ast & 1& \ast \\ -1& 1& \ast & 3& \ast \\ 2& 0& \ast & 1& \ast \end{array}\right],R=\left[\begin{array}{lllll}1& 0& 1& 0& 3\\ 0& 1& 2& 0& 1\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 0\end{array}\right]$$

(A) Find a basis for each of the four fundamental subspaces of $A$

(B) Find the third column of matrix $A$

四

Let

$$A=\left[\begin{array}{ccc}1& -1& -1\\ 2& a& 1\\ -1& 1& a\end{array}\right],B=\left[\begin{array}{cc}2& 2\\ 1& a\\ -a-1& -2\end{array}\right]$$

For which valus(s) of $a$, the matrix equation $AX=B$ has no solution, a unique solution, or infinitely many solutions? Where $X$ is a $3\times 2$ matrix. Solve $AX=B$ if it has at least one solution

五

Let $M_{2\times 2}(\mathbb{R})$ be the vector space of $2\times 2$ real matrices. Let

$$\begin{array}{rl}& A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],B=\left[\begin{array}{ll}0& 1\\ 0& 0\end{array}\right],C=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]\end{array}$$

Consider the map

$$T:M_{2\times 2}(R)\to R^3,T(X)=\left[\begin{array}{l}\mathrm{tr}(A^{\mathrm{\top }}X)\\ \mathrm{tr}(B^{\mathrm{\top }}X)\\ \mathrm{tr}(C^{\mathrm{\top }}X)\end{array}\right]$$

for any $2\times 2$ real matrix $X$, where $\mathrm{tr}(D)$ denotes the trace of a matrix $D$

The trace of an $n\times n$ matrix $D$ is defined to be the sum of all the diagonal entries of $D$, in other words,

$$\mathrm{tr}(D)=d_{11}+d_{12}+\cdots +d_{nn}$$

(A) Show that $T$ is a linear transformation

(B) Find the matrix representation of $T$ with respect to the ordered basis

$$v_1=\left[\begin{array}{ll}1& 0\\ 0& 0\end{array}\right],v_2=\left[\begin{array}{ll}0& 0\\ 0& 1\end{array}\right],v_3=\left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right],v_4=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$$

for $M_{2\times 2}(R)$ and the standard basis

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

for $R^3$

(C) Can we find a matrix $X$ such that $T(X)=\left[\begin{array}{c}1\\ -2\\ 1\end{array}\right]$ ? If yes, please find one such matrix

otherwise, give an explanation

六

Let $A$ be an $m\times n$ matrix, $B$ be an $m\times p$ matrix, and $C$ be an $q\times p$ matrix. Show that

$$\mathrm{rank}\left[\begin{array}{ll}A& B\\ 0& C\end{array}\right]\ge \mathrm{rank}A+\mathrm{rank}C\text{. }$$

where 0 is the $q\times n$ zero matrix