D. Determinants, Jordan form, Euclidean structures.
\begin{enumerate}
\item
If $A$ and $B$ are linear transformations such that
$AB=0$, $A\neq0$, $B\neq0$, then
$\det A=\det B=0$.
\item
Prove the following and see if the converse is true.
\begin{enumerate}
\item
If $A$ and $B$ are similar, then $\det A=\det B$.
\item
If $A$ and $B$ are similar, then $A$ and $B$ have the
same characteristic polynomial.
\item
If $A$ and $B$ have the same characteristic polynomial,
then $\det A=\det B$.
\end{enumerate}
\item On \(\mathcal{P}_n\), the space of polynomials \(p(t)\) of degree \(\leq n\), 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. \[
\begin{pmatrix}
0&1&\alpha \\ 0&0&1 \\ 0&0&0
\end{pmatrix}
\quad
\begin{pmatrix}
0&1&0 \\ 0&0&1 \\ 0&0&0
\end{pmatrix}
\]
\item For \(x=(x_1,x_2)\in \mathbb{R}^2\) define \[ \|x\|:=\max \{ |x_1|, |x_2| \}. \] Check that this function satisfy triangle inequality. And prove that there is no innerproduct \((~,~)\) in \(\mathbb{R}^2\) so that \(\|x\|^2=(x,x)\).
\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}