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\begin{document}


\begincenter{Peter Cholak\\Math 221\\Quiz\\Nov. 25, 1996}\endcenter


\heading

Work in $\mathbb{R}^4$ with the Euclidean inner product.
Consider the two vectors $\vec{u} = (1,1,1,0)$ and $\vec{v}
=(-1,1,0,1)$; both of whose norm is $\sqrt{3}$.  Let $W$ be the
space spanned by these two vectors.  (Be sure to show all needed
work.)
\\
\begin{problem}\label{oth}
  (3 points) Are $\vec{u}$ and $\vec{v}$ orthogonal?
\end{problem}
\vspace{5cm}
\begin{problem}\label{basis}
  (8 points) Find a basis for $W^\bot$.
\end{problem}
\newpage
\begin{problem}
  (3 points) Assume $A = \begin{bmatrix} \vec{u} \\ \vec{v}
  \end{bmatrix}$ and $A\vec{x}_0 = \vec{b}$. Find the general
  solution to $A\vec{x} = \vec{b}$ (in terms of $\vec{x}_0$).
  (Hint: use the answer to the last problem.)
\end{problem}
\vspace{4cm}
\begin{problem}
  (3 points) Use the answer to problem \ref{basis} to find a
  basis for $\mathbb{R}^4$ that includes $\vec{u}$ and $\vec{v}$.
\end{problem}
\vspace{4cm}
\begin{problem}
  (8 points) Write $\vec{w} = (0,2,4,1)$ as the $\vec{w}_1 +
  \vec{w}_2$, where $\vec{w}_1$ is in $W$ and $\vec{w}_2$ is in
  $W^\bot$. (Hint: The answer to \ref{oth} will be useful here
  and the fact that the norm of $\vec{u}$ and $\vec{v}$ is
  $\sqrt{3}$.)
\end{problem}

\end{document}

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