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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due February 5
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf MthSc 853 $|$ Linear
    Algebra $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}

\item Let $\mathcal{P}_2$ be the vector space of all polynomials
  $p(x)=a_0+a_1x+a_2x^2$ over $\R$, with degree $\leq 2$. Let
  $\xi_1,\xi_2,\xi_3$ be distinct real numbers, and define
  \[
  \ell_j=p(\xi_j)\quad\mbox{for}\quad j=1,2,3\,.
  \]
  \begin{enumerate}
  \item Show that $\ell_1,\ell_2,\ell_3$ are linearly independent
    functions on $\mathcal{P}_2$.
  \item Show that $\ell_1,\ell_2,\ell_3$ is a basis for the dual space
    $\mathcal{P}'_2$.
  \item Find polynomials $p_1(x),p_2(x),p_3(x)$ in $\mathcal{P}_2$ of
    which $\ell_1,\ell_2,\ell_3$ is the dual basis in
    $\mathcal{P}'_2$.
  \end{enumerate}



\item Let $W$ be the subspace of $\R^4$ spanned by $(1,0,-1,2)$ and
  $(2,3,1,1)$. Which linear functions
  $\ell(x)=c_1x_1+c_2x_2+c_3x_3+c_4x_4$ are in the annihilator of $W$?
  Write your answer by giving an explicit basis of $W^\perp$. 


  
\item Let $T\colon X\to U$ be a linear map. Prove the following:
  \begin{enumerate}
  \item The image of a subspace of $X$ is a subspace of $U$.
  \item The inverse image of a subspace of $U$ is a subspace of $X$.
  \end{enumerate}


\item Prove Theorem 3.3 in Lax:
\begin{enumerate}
  \item The composite of linear mappings is also a linear mapping. 
  \item Composition is distributive with respect to the addition of
    linear maps, that is,
    \[
    (R+S)\circ T=R\circ T+S\circ T
    \]
    and 
    \[
    S\circ(T+P)=S\circ T+S\circ P\,,
    \]
    where $R,S\colon U\to V$ and $P,T\colon X\to U$.  
\end{enumerate}



\item Let $X$ be a finite-dimensional vector space over $K$ and let
  $\{x_1,\dots,x_n\}$ be an ordered basis for $X$. Let $U$ be a vector
  space over the same field $K$ but possibly with a different
  dimension, and let $\{u_1,\dots,u_n\}$ be an arbitrary set of
  vectors in $U$. Show that there is precisely one linear
  transformation $T\colon X\to U$ such that $Tx_i=u_i$ for each
  $i=1,\dots,n$.



\item Let $X$ and $U$ be vector spaces, and suppose that $Y$ is a
  subspace of $X$. Let $Q\colon X\to X/Y$ be the canonical quotient
  map sending $x\stackrel{Q}{\longmapsto}\{x\}$, and let $T\colon
  X\to U$ be a linear map. Give necessary and sufficient conditions
  for the existence of a unique linear map $S\colon X/Y\to U$ such
  that $T=S\circ Q$. When this happens, the map $T$ is said to \emph{factor
    through} the quotient space, as shown by the following commutative
  diagram:
  \[
  \xymatrix{X\ar[rr]^T\ar[dr]_Q && U \\ & X/Y\ar@{-->}[ur]_S}
  \]
  Prove all of your claims.


\end{enumerate}

\end{document}
