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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 3
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2026 $|$ M.~Macauley}}


\begin{document}

%$\;$

\begin{enumerate}
  
  %%------------------------------------------------------------------------

\item Let $x_1,x_2$ be a basis of $X=\R^2$, and $\ell_1,\ell_2$ 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. \medskip
  \end{enumerate}
  
  %%------------------------------------------------------------------------
  
\item Let $A\colon X\to U$ be a linear map between finite-dimensional
  inner product spaces, and let $A^*\colon U\to X$ denote the
  adjoint. Prove each of the following equalities:
  \begin{multicols}{2}
    \begin{enumerate}
    \item $N_{A^*}=R_A^\perp$
    \item $R_{A^*}=N_A^\perp$
    \item $N_A=R_{A^*}^\perp$
    \item $R_A=N_{A^*}^\perp$. \medskip
    \end{enumerate}
  \end{multicols}
  
  %%------------------------------------------------------------------------ 
  
\item Let $A\colon X\to U$ be a linear map between finite-dimensional
  inner product spaces. Prove that $A$ maps $R_{A^*}$ bijectively onto
  $R_A$. \medskip

 %%------------------------------------------------------------------------
  
\item The map $A\colon X\to U$ between finite-dimensional vector spaces has a \emph{left inverse} if there is a linear map $L\colon U\to X$ such that $LA=I_X$, the identity on $X$. It has a \emph{right inverse} if there is a linear map $R\colon U\to X$ such that $AR=I_U$ is the identity on $U$.
  \begin{enumerate}
  \item Show that if $A$ has a left inverse, then $Ax=u$ has \emph{at
    most} one solution. Give a condition on $u$ that completely
    characterizes when there is a solution.
  \item Show that if $A$ has a right inverse, then $Ax=u$ has \emph{at
    least} one solution. If $Ax_p=u$ for some particular $x_p\in X$,
    then describe all solutions for $x$ in this case. What condition
    ensures that there will be \emph{only} one solution?
  \item What are the possibilities for the rank of $A$ if it has
    a left inverse? What if it has a right inverse? \medskip
  \end{enumerate}
  
  %%------------------------------------------------------------------------
  
\item Show that if $P\colon X\to X$ is a projection (i.e., $P^2=P$),
  then $X=R_P\oplus N_P$. \medskip
  
  %%------------------------------------------------------------------------
  
\item Show that if $P\colon X\to X$ is a projection, then
  $R_P^\perp=N_P$ if and only if $P$ is self-adjoint. \medskip
  
  %%------------------------------------------------------------------------
  
\item Consider four data points $(0,0)$, $(1,8)$, $(3,8)$, and
  $(4,20)$. In this problem, we will find the best fit line $C+Dt$
  through these points, using least squares.
  \begin{enumerate}
  \item Write down an equation in matrix form, $Ax=b$, where
    $x=(C,D)^T$, that has no solution because these four points are
    not co-linear.
  \item Find the best fit line by solving the related equation
    $A\hat{x}=p$, where $p$ is the orthogonal projection of $b$ onto
    the range of $A$.
  \item Repeat the previous steps to find the best fit parabola
    $C+Dt+Et^2$. 
  \end{enumerate}

    %%------------------------------------------------------------------------
  
\end{enumerate}

\end{document}
