숙제 5

D. Determinants, Jordan form, Euclidean structures.

\begin{enumerate}
\item
If $A$ and $B$ are linear transformations such that
$AB=0$, $A\ne 0$, $B\ne 0$, then
$detA=detB=0$.

\item
Prove the following and see if the converse is true.
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\item On ${\mathcal{P}}_n$, the space of polynomials $p(t)$ of degree $\le n$, consider the differential operator $D=\partial /\partial t$. Find the Jordan form of $D$. Find the minimal polynomial of $D$. (Hint: Use the standard basis $1,t,t^2,\dots ,t^n$ to compute the matrix representation for $D^k$.)

\item Read the theorem 12, p.{} 57 of our textbook.(It is not necessary to read the proof.) Then show that the following matrices are similar. \[

$$\left(\begin{array}{ccc}0& 1& \alpha \\ 0& 0& 1\\ 0& 0& 0\end{array}\right)$$

\quad

$$\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right)$$

\]

\item For $x=(x_1,x_2)\in {\mathbb{R}}^2$ define $$\Vert x\Vert :=max\{|x_1|,|x_2|\}.$$ Check that this function satisfy triangle inequality. And prove that there is no innerproduct $(,)$ in ${\mathbb{R}}^2$ so that $\Vert x{\Vert }^2=(x,x)$.

\item Let $A$ be a self-adjoint linear transformation. Show that the following formula holds: $$(Ax,y)+(Ay,x)=(A(x+y),(x+y))–(Ax,x)–(Ay,y).$$ Use this formula to show that, if $(Ax,x)=0$ for all $x$ then $A=0$.

\end{enumerate}