2022秋 线代期末试巻(回忆版)

1. Multiple Choice. Only one choice is correct

1-(1)

Let $A,B,C$ be $n\times n$ matrices with $B$ invertible and $AB=C$ . Which of the following must be true？

(A) The row spaces of $A$ and $C$ are the same

(B) The null spaces of $A$ and $C$ are the same

(C) The column spaces of $A$ and $C$ are the same

(D) The determinants of $A$ and $C$ are the same

1-(2)

Let $P$ be a $5\times 5$ permutation matrix. Which of the following is false？

(A) $P$ is an orthogonal matrix

(B) $P$ must have real eigenvectors

(C) There always exists an invertible real matrix $Q$ such that $Q^{-1}PQ$ is diagonal

(D) The equation $P_x=0$ has only zero solution

1-(3)

Let $A$ be an $n\times n$ real sysmmetric matrix. Which of the following statements must be true？

(A) A must have $n$ distinct eigenvalues

(B) Some of the complex eigenvalues of $A$ need not be real

(C) Any $n$ linearly independent eigenvalues eigenvectors of $A$ are pairwise or thogonal

(D) There is an athogonal matrix $Q$, such that $Q^{\mathrm{\top }}AQ$ is diagonal

1-(4)

Let ${\alpha }_1=\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right],{\alpha }_2=\left[\begin{array}{l}2\\ 1\\ 1\end{array}\right],{\beta }_1=\left[\begin{array}{l}2\\ 5\\ 9\end{array}\right],{\beta }_2=\left[\begin{array}{l}1\\ 0\\ 1\end{array}\right]$

If $\gamma$ can be written as a linear combination of ${\alpha }_1,{\alpha }_2$, and $\gamma$ can also be written as a linear combination of ${\beta }_1,{\beta }_2$, then $r$ has the form

(A) $k\left[\begin{array}{l}1\\ 5\\ 8\end{array}\right],k\in R$

(B) $k\left[\begin{array}{l}3\\ 5\\ 10\end{array}\right],k\in R$

(C) $k\left[\begin{array}{c}-1\\ 1\\ 2\end{array}\right],k\in R$

(D) $k\left[\begin{array}{l}3\\ 3\\ 4\end{array}\right],k\in R$

1-(5)

Which of the following matrices is congruent to the identity matrix？

(A)

$$\left[\begin{array}{lll}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$$

(B)

$$\left[\begin{array}{lll}1& 2& 1\\ 2& 7& 1\\ 1& 1& 8\end{array}\right]$$

(C)

$$\left[\begin{array}{ccc}2& -1& 2\\ -1& 3& -3\\ 2& -3& -4\end{array}\right]$$

(D)

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

2. Fill in the blanks

2-(1)

Let $A$ be a $2\times 2$ matrix, which has two linearly independent eigenvectors $v_1$ and $v_2$ such that $A^2(v_1-v_2)=2v_1+v_2$ . Then $\det \left(A^4\right)=\underset{―}{}$

2-(2)

The singular values of the matrix $A=\left[\begin{array}{ll}3& 2\\ 2& 3\\ 2& -2\end{array}\right]$ are $\underset{―}{}$

2-(3)

Let $A$ be a $3\times 3$ matrix which has eigenvalues $-1,0,1$. Suppose that $(A+a{I}_3)A(A$ $b{I}_3)=0$, where $I_3$ is the $3\times 3$ identity matrix. Then $a=$ $\underset{―}{}$ , $b=$ $\underset{―}{}$

2-(4)

If $A=\left[\begin{array}{ccc}0& 0& 1\\ x& 1& 2x-3\\ 1& 0& 0\end{array}\right]$ is diagonalizable, then $x=\underset{―}{}$

2-(5)

Let $A$ be a $4\times 4$ symmetric matrix such that $A^2+A=0$. Suppose that $A$ has rank $3$

A diagonal matrix that is similar to $A$ is $\underset{―}{}$

3

Let $A_n$ be the $n\times n$ matrix

$$A_n=\left[\begin{array}{cccccc}1& -a& 0& 0& ...& 0\\ -a& 1& -a& 0& ...& 0\\ 0& -a& 1& -a& \cdots & 0\\ ⋮& & \ddots & \ddots & \ddots & ⋮\\ 0& 0& \cdots & -a& 1& -a\\ 0& 0& \cdots & 0& -a& 1\end{array}\right]\text{. }$$

(A) Find constants $b,c$ such that the sequence $\det \left(A_n\right)$ satisfies

$$\det \left(A_n\right)=b\cdot \det \left(A_{n-1}\right)+c\cdot \det \left(A_{n-2}\right)\text{ for all }n⩾3\text{. }$$

(B) Find a matrix $B$ such that $x_n=B{x}_{n-1}$ for $n⩾3$, where $x_n=\left[\begin{array}{l}\det \left(A_n\right)\\ \det \left(A_{n-1}\right)\end{array}\right]$

(C) For $a^2=\frac{3}{16}$, find an expression for $\det \left(A_n\right)$ for all $n⩾3$

4

Suppose $\alpha ,\theta \in (0,\frac{\pi }{2})$

(A) Compute $A_n={\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]}^n\left[\begin{array}{ll}9& 0\\ 0& 4\end{array}\right]{\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]}^n$ for all $n⩾1$

(B) Find a singular value decomposition (SVD) of $A_n$ for each $n⩾1$

(C) Show that the matrix $A_1$ is symmetric if and only if $\alpha =\theta$

(Hint: the formula $\sin (\theta -\alpha )=\sin \theta \cos \alpha -\cos \theta \sin \alpha$ may be useful. )

(D) Prove that if $A_1$ is symmetric, then $A_n$ is positive definite for every $n⩾1$

5

Consider the quadratic form $f(x_1,x_2,x_3)=x_1^2+2x_2^2-2x_3^2-4x_1{x}_3$

(A) Find the symmetric matrix $A$ such that $f(x)=x^{\mathrm{\top }}Ax$ for all $x={(x_1,x_2,x_3)}^{\mathrm{\top }}$, and find an orthogonal matrix $Q$ such that $Q^{\mathrm{\top }}AQ$ is a diagonal matrix

(B) The quadric surface defined by the equation $f(x,y,z)=2023$ is $\underset{―}{}$

• (A) a hyperboloid of one sheet
• (B) a hyperboloid of two sheets
• (C) an ellipsoid
• (D) none of the above

6

For any $a={(a_1,\cdots ,a_n)}^{\mathrm{\top }}\in R^n$, put $\parallel a\parallel =\sqrt{a_1^2+\cdots +a_n^2}$. Let $x,y\in R^n$ be nonzero vectors

(A) Show that if there is an orthogonal matrix $S$ such that $Sx=y$, then $\parallel x\parallel =\parallel y\parallel$

(B) Let $N$ be the null space $N\left(x^{\mathrm{\top }}\right)$ of the $1\times n$ matrix $x^{\mathrm{\top }}$. Show that $\dim N=n-1$

(C) Let ${\alpha }_2,\cdots ,{\alpha }_n$ be a basis of $N$. Show that the system ${\alpha }_1:=x,{\alpha }_2,\cdots {\alpha }_n$ is linearly indindegende independent

(D) Let $A$ be the matrix with ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n$ as its columns. Let $A=QR$ be a factorization with $Q$ orthogonal and $R$ upper triangular. Write $R=\left(r_{ij}\right)$. Show that $\left|r_{11}\right|=\parallel x\parallel$

(e) Prove that if $\parallel x\parallel =\parallel y\parallel$, then there exists an orthogonal matrix $S$ such that $Sx=y$