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\documentclass{article}
\usepackage{amsmath,amssymb}

\input{ExamMacros.tex}

\prof{Peter Cholak}
%\prof{George McNinch}
\class{Math 221}
\quizdate{Wednesday, October 16}
\examname{Quiz 2}
\begin{document}

%\pagestyle{empty}

\heading

\noindent Be sure to justify your answers! \\

\begin{problem}
  (7 points) Find a vector $\vec{u}$ in $\mathbb{R}^2$ so that
  $\vec{u}$ is orthogonal to $\vec{v} = (1,1)$, and so that
  $\|\vec{u}\|=2$.
\end{problem}

\vfill

%\begin{problem}
%  $T$ is a linear transformation such that $T(\vec{u})= (2,1,3)$
%  and $T(\vec{v})= (1,1,1)$.  Compute $T(2\vec{v}-3\vec{u})$.
%\end{problem}

%\vfill

\begin{problem}
  (6 points) Which of the followings are subspaces of the vector
  space of $2\times2$ matrices?

  \begin{subproblem}
    Matrices of the form $\begin{bmatrix}
      a & 2a \\
      b &  0\\
               \end{bmatrix}$, for any real numbers $a$ and $b$.
  \end{subproblem}
\vspace{3cm}
   \begin{subproblem}
     Matrices of the form $\begin{bmatrix}
       a+1 & 2a \\
       b &  0\\
               \end{bmatrix}$, for any real numbers $a$ and $b$.
  \end{subproblem}
\end{problem}

\vfill

\noindent  {\em Turn over; more problems on the back!}
\newpage


\begin{problem}
  (6 points) $T$ is the function from $\mathbb{R}^2$ to
  $\mathbb{R}^2$ such that $T(x,y) = (2x-y,y)$.  Is $T$ linear?
  Is $T$ one-to-one?  What is the range of $T$?
\end{problem}

\vfill

\begin{problem}
  (6 points) Consider the vector space $P$ of polynomial
  functions.  Is $2x-x^2$ in span of $1-x, 3x-x^2$? Is $3$ in the
  span of $1-x, 3x-x^2$?
\end{problem}

\vfill
% another problem from George.
%\begin{problem}
%  Let $T$ be the linear transformation with standard matrix
%  $\begin{bmatrix}
%    4 & 1 \\
%    0 & 2 \\
%  \end{bmatrix}$.
%  Decide which of the vectors $\vec{u}$ and $\vec{v}$ are eigenvectors for $T$. 
% For the eigenvectors that you find, indicate the corresponding eigenvalue.
%  $$ \vec{u} = (1,2) \qquad \vec{v} = (3,0)$$
%\end{problem}
 

\end{document}





