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\begin{document}

Your name:
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Instructor (please circle): 
\hspace{1in}
Alex Barnett
\hskip .7in
Naomi Tanabe
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{\bf \large Math 22 Fall 2016, Homework 2, due Wed Sep 28}

{\em Please show your work, and check your answers. No credit is given for solutions without work or justification.}

\vskip .05in

\ben

\item
Consider the set of three vectors
$\mbf{v}_1=\left[\begin{array}{r}1\\0\\-1\end{array}\right]$,
$\mbf{v}_2=\left[\begin{array}{r}2\\3\\4\end{array}\right]$,
$\mbf{v}_3=\left[\begin{array}{r}1\\6\\h\end{array}\right]$.
\ben
\item With $h=0$, is the set linearly independent, and why?
If not, give a dependence relation between $\mbf{v}_1$, $\mbf{v}_2$ and $\mbf{v}_3$.

\vspace{3in}

\item With $h=11$, is the set linearly independent, and why?
If not, give a dependence relation between $\mbf{v}_1$, $\mbf{v}_2$ and $\mbf{v}_3$.




%lin xform a bit?

\een

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% 7 pts
\item 
Consider the following network flow problem, where the numbers indicate
known flows, and $x_1,\ldots,x_5$ unknown flows, on their respective edges:
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\ben

\item Find the {\em reduced} echelon form for the linear system
corresponding to the flow problem:

\vspace{3in}

\item If a solution exists,
write the {\em parametric vector form} of the solution set
(include the numerical values of any vectors you use):



\vspace{2in}



\item [BONUS:] In a general such network flow problem (where all flows
to the outside world are known, and all interior flows unknown),
what {\em geometric property of the network} does 
the number of free variables tell you? Why?

\een


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% 6 pts
\item Let $\mbf{v}_1, \ldots, \mbf{v}_n$ be a set of $n$ vectors
in $\RR^m$.

\ben
\item

Define Span$\{\mbf{v}_1, \ldots,\mbf{v}_n\}$,
using clear verbal or mathematical language. (Careful:
do not assume later theorems or properties of span!)
% testing 1.3 (HW1)

\vspace{2.5in}

\item If $n<m$, can it be that Span$\{\mbf{v}_1, \ldots,\mbf{v}_n\} = \RR^m$ ?
Prove your answer (ie explain as clearly as you can).
% testing 1.4

\vspace{2.5in}

\item If $n=m$ and Span$\{\mbf{v}_1, \ldots,\mbf{v}_n\} = \RR^m$,
what can you say about linear (in)dependence of the set
$\{\mbf{v}_1, \ldots,\mbf{v}_n\}$ ?
Prove your answer.
% testing 1.7


\een

\een

\end{document}
