2021秋 线性代数期末

一

Label the following statements as True or False. Along with your answer, provide an informal proof, counterexample, or other explanation

1-a

The sum of two positive operators on a finite-dimensional complex inner product space is positive

1-b

Let $V$ be a 5-dimensional vector space and $T\in \mathcal{L}$. Then there exists a 4-dimensional subspace $U$ of $V$ invariant under $T$

1-c

Any polynomial of degree $n$ with leading coefficients $(-1{)}^n$ is the characteristic polynomial of some linear operator

1-d

If $x,y$ and $z$ are vectors in an inner product space such that $⟨x,y⟩=⟨x,z⟩$, then $y=z$

1-e

Every normal operator is diagonalizable

二

Suppose $T\in \mathcal{L}(R^3)$ is defined by

$$T(x_1,x_2,x_3)=(2x_1,x_2-x_3,x_2+x_3)$$

2-a

Determine the eigenspace of $T$ corresponding to each eigenvalue

2-b

Find the Jordan form and a Jordan basis of $T$

2-c

Find the minimal polynomial of $T$

2-d

Find the trace of $T$, trace $T$

2-e

Find the determinant of $T$, det $T$

三

Suppose $V$ is a finite-dimensional inner product space, $T\in \mathcal{L}(V)$ is normal, and $U$ is a subspace of $V$ that is invariant under $T$ . Show that $U^{\perp }$ is invariant under $T$

四

Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $v$ be a nonzero vector in $V$ . The subspace

$$U=\mathrm{span}(\{v,Tv,T^{2}v,\cdots \})$$

is called the T-cyclic subspace of $V$ generated by $v$

4-a

Show that $U$ is a finit-dimensional invariant subspace of $V$

4-b

Let $k=\dim U$ . Show that $\{v,T_v,T_v^2,\cdots ,T^{k-1}v\}$ is a basis for $U$

4-c

If $a_{0}v+a_{1}Tv+a_2{T}^{2}v+\cdots +a_{k-1}{T}^{k-1}(v)+T^k(v)=0$, show that the characteristic polynomial of $T{|}_v$ is

$$f(t)=(-1{)}^k(a_0+a_{1}t+\cdots +a_{k-1}{t}^{k-1}+t^k)$$

4-d

Let $g(t)$ be the characteristic polynomial of $T$, show that $g(T)=0$, where 0 is the zero operator. That is, $T$ "satisfies" its characteristic equation

五

If $F=C$, show that $T$ is an isometry if and only if $T$ is normal and $|\lambda |=1$ for every eigenvalue $\lambda$ of $T$

六

Let $P_2(R)$ and $P_1(R)$ be the polynomial spaces with inner products defined by

$$⟨f,g⟩={\int }_{-1}^{1}f(x)g(x)dx,f,g\in P_2(R)$$

Let $T:P_2(R)\to P_1(R)$ be the linear operator defined by

$$T(f(x))=f^{\prime }(x)$$

6-a

Find orthonormal bases $\{v_1,v_2,v_3\}$ for $P_2(R)$ and $\{u_1,u_2\}$ for $P_1(R)$

6-b

Find $p\in P_1(R)$ that makes ${\int }_{-1}^1|x^5-p(x){|}^{2}dx$

as small as possible

6-c

Find the singular values ${\sigma }_1,{\sigma }_2$ of $T$ such that $T({\nu }_i)={\sigma }_i{u}_i,i=1,2$, and $T({\nu }_3)=0$

七

Let $V$ be a real inner product space. A function $f:V\to V$ is called a rigid motion if

$$\parallel f(x)-f(y)\parallel =\parallel x-y\parallel$$

for all $x,y\in V$. For example, any isometry on a finite-dimensional real inner product space is a rigid motion. Another class of rigid motions are the translations.

A function $g:V\to V$, where $V$ is a real inner product space, is called a translation if there exists a vector $v_0\in V$ such that $g(v)=v+v_0$ for all $v\in V$. Let $f:V\to V$ be a rigid motion on a finite-dimensional real inner product space $V$, show that there exists a unique isometry $T$ on $V$ and a unique translation $g$ on $V$ such that $f=g\circ T$