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\begin{document}
\centerline{Math 201 Homework for Week 2, Friday}
\bigskip

\begin{prob}
  Let $V=\R^2$. For the following pairs of operations, decide whether they make
  $V$ into a vector space over $\R$. Justify your answer. In what follows, let
  $(x_1,x_2),(y_1,y_2) \in \R^2$ and $r\in \R$.
  \begin{enumerate}
    \item \[(x_1,x_2)+(y_1,y_2)=(x_1+y_1,x_2y_2) \quad \text{and} \quad r\cdot(x_1,x_2)=(rx_1,x_2).\] 
    \item \[(x_1,x_2)+(y_1,y_2)=(2x_1+2y_1,3x_2+3y_2) \quad \text{and} \quad r\cdot(x_1,x_2)=(rx_1,rx_2).\] 
  \end{enumerate}
\end{prob} 
 
\begin{prob}
  Here are two templates for showing a subset~$W$ of a vector space~$V$
  over a field~$F$ is a subspace:
  \medskip

  {\bf Proof 1.} First note that~$\vec{0}\in W$ since \underline{\hspace{2cm}}.
  Hence, $W\neq\emptyset$.
  Next, suppose that~$u,v\in W$. Then \underline{\hspace{2cm}}.
  Hence,~$u+v\in W$.  Now suppose~$r\in F$ and~$w\in W$.  Then
  \underline{\hspace{2cm}}.  Therefore,~$r\cdot w\in W$.  
  \hfill$\Box$
  \medskip

  {\bf Proof 2.} First note that~$\vec{0}\in W$ since \underline{\hspace{2cm}}.
  Hence, $W\neq\emptyset$.
  Next, suppose that~$r\in F$ and~$u,v\in W$. Then \underline{\hspace{2cm}}.
  Hence,~$ u+r\cdot v\in W$.
  \hfill$\Box$

  Use one of these two templates for each of the following
  exercises.
  \begin{enumerate}
    \item Show that~$W=\{(x,y,z)\in\R^3\mid 2x-y-3z=0\}$ is a subspace of~$\R^3$.
    \item Show that the set $W=\left\{ f\colon\R\to\R: f(t)=f(-t) \right\}$
      is a subspace of the vector space of real-valued functions of one variable.
      (You will need to carefully use the definitions given in Example~1.12, p.~84, of the text.)
  \end{enumerate}
\end{prob}
\end{document}
