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\lhead{{\sf Bridge course (Algebra)}: Friday July 26}
\chead{}\rhead{\thepage}
\lfoot{Due Monday, July 29 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

\noindent \textbf{Topic}: Adjoints. \\

\noindent \textbf{Do}: Answer the following questions. \\

\begin{enumerate}

%%------------------------------------------------------------------------

 %%--------------------------------------------------------------------- 

\item Let $A\colon X\to U$ be a linear map between finite-dimensional inner product spaces. Recall that its \emph{adjoint} is a linear map $A^*\colon U\to X$ such that $(Ax,u)=(x,A^*u)$ for all $x\in X$ and $u\in U$. 
\begin{enumerate}
    \item Give a direct proof that $R_A^\perp=N_{A^*}$, by showing that these are equal as sets. 
    \item Establish the following equalities as simple corollaries:
    \begin{multicols}{3}
    \begin{enumerate}[(i)]
    \item $R_{A^*}=N_A^\perp$
    \item $N_A=R_{A^*}^\perp$
    \item $R_A=N_{A^*}^\perp$.
    \end{enumerate}
    \end{multicols}
    \item Show that $A$ maps $R_{A^*}$ bijectively onto $R_A$. \\
\end{enumerate}


 %%------------------------------------------------------------------------ 

\item Let $x_1,x_2$ be a basis of $X=\R^2$, and $\ell_1,\ell_2\in X'$ the
  dual basis. Carry out the steps below for the linear map $A\colon
  X\to X$ defined by $A(x_1)=x_1$ and $A(x_2)=x_1+x_2$ with respect to
  the standard dot product, and then with respect to each of the following inner
  products:
  \[
  \<x,y\>:=\begin{bmatrix}y_1&y_2\end{bmatrix}\begin{bmatrix}2
    & 1 \\ 1 & 2\end{bmatrix}\vv{x_1}{x_2},\qquad\text{and}\qquad
    \<x,y\>:=\begin{bmatrix}y_1&y_2\end{bmatrix}\begin{bmatrix}2
      & -1 \\ -1 & 2\end{bmatrix}\vv{x_1}{x_2}.
      \]
  \begin{enumerate}
  \item Find $v_i\in X$ for which $\ell_i=\<-,v_i\>$, for $i=1,2$.
  \item Find $y_i\in X$ for which $A'(\ell_i)=\ell_i\circ
    A=\<-,y_i\>$, for $i=1,2$.
  \item Find the adjoint $A^*\colon X\to X$ with respect to this inner
    product. \bigskip
  \end{enumerate}  

%%------------------------------------------------------------------------ 

%%------------------------------------------------------------------------ 

\end{enumerate}

\end{document}
