% LaTeX file -- M245, Fall 2012, Supplement on Duality

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large Math 245\\
Supplementary problems on duality}
\end{center}
\vskip 1em 

\begin{Ex}
\item  \begin{parts}
\item Let $v_1=(1,0,-1)^t$, $v_2=(1,1,1)^t$, and $v_3=(2,2,0)^t$. This is
a basis for $\bR^3$. Find the corresponding dual basis.\\
\hint Find the inverse of the matrix $[v_1,v_2,v_3]$.

\item Let $V$ be the vector space of polynomials of degree 2. Consider
the basis for $V'$ given by:
\[ \phi_i(p)=\int_0^i p(x)\,dx \qfor i=1,2,3 .\]
Find a basis for $V$ dual to $\{\phi_1,\phi_2,\phi_3\}$.
\end{parts}


\item 
\begin{parts}
\item 
 Fix $A\in\L(V)$. Define $T\in\L(\L(V))$ by $T(B)=AB-BA$. Let
$\tau$ be the trace functional on $\L(V)$ given by $\tau(B)={\rm trace}(B)$.
Compute $T^t(\tau)$.
\item  Let $\phi$ be a functional in $\L(V)'$. Show that there is a unique
transformation $A$ in $\L(V)$ so that $\phi(B)=\tau(AB)$ for all $B\in\L(V)$.
\hint consider $\phi(E_{ij})$.
\end{parts}

\item 
\begin{parts}
\item 
Show that $E_{ij}$ and $E_{ii}-E_{jj}$ are {\it commutators}, meaning they are
of the form $AB-BA$, for $i\neq j$.
\item Hence show that the linear span of all commutators consists of all
matrices of trace zero.
\item Hence show that if $\phi\in\L(V)'$ satisfies $\phi(AB-BA)=0$ for all $A$ and
$B$ in $\L(V)$, then $F$ is a multiple of $\tau$, the trace functional.
\end{parts}

\item Let $T\in\L(V)$. Let $\rho_T$ be the linear map from $\bF[x]$ to $\L(V)$
given by $\rho_T(p) = p(T)$.
Explain the factorization of $\rho_T$ through the quotient by the kernel of
$\rho_T$ to the range of $\rho_T$ and then into $\L(V)$ using the language
developed to study linear transformations. \\
(\textit{Note: thinking of $\rho_T$
as a linear map hides the fact that it is also multiplicative.})

\item Let $E\in\L(V)$ be an idempotent with range $M$ and kernel $N$.
\begin{parts}
 \item Explain the factorization of $E$ through the quotient by the kernel, 
 to the range, and then back into $V$.
 \item Explain what happens when you take the transpose of this factorization.
 In particular, what is the range of $E^t$?
\end{parts}

\end{Ex}
\end{document}
