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\begin{document}

Your name:
\vskip 12pt

Instructor (please circle): 
\hspace{1in}
Alex Barnett
\hskip .7in
Naomi Tanabe
\vskip 12pt

{\bf \large Math 22 Fall 2016, Homework 1, due Wed Sep 21}

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

\vskip .25in

\ben

\item
Given the system of equations, answer the following questions.
\bea
x_2 + 2x_3 &=& -1 \nonumber \\
2x_1-x_2 \quad &=& 3 \nonumber \\
x_1 +x_2+3x_3 &=& 0 \nonumber
\eea
% note I deliberately didn't line them up

\ben
\item Write the augmented matrix and transform it to reduced echelon form:

\vspace{4in}

\item Write the {\em general} solution to the linear system
(by any non-computer method), if there is one:
\een

\newpage %%%%%%%%%%%%%%%%

\item 
For what value(s) of $h$ is the vector
$\left[\begin{array}{r}1\\0\\h\end{array}\right]$
in Span$\left\{
\left[\begin{array}{r}8\\0\\-4\end{array}\right],\,
\left[\begin{array}{r}-2\\1\\1\end{array}\right],\,
\left[\begin{array}{r}4\\2\\-2\end{array}\right]
\right\}$ ?

\vspace{6in}

For the value(s) of $h$ when the vector is in the span,
is the set of weights in the linear combination which produces the vector
unique? Explain.


\newpage %%%%%%%%%%%%%%%%


\item
True or false (no working needed, just circle the answer):
% will be one pt each

\vg

\ben
\itemsep 10ex

\item T / F: \qquad
\parbox{4in}{
A linear system with less equations than unknowns can have a unique
solution.
}

\item T / F: \qquad
A linear system with less equations than unknowns can be consistent.

\item T / F: \qquad
\parbox{4in}{
If $\mbf{u}=(1,0)$ and $\mbf{v}=(0,1)$
then Span$\{\mbf{u},\mbf{v}\}$
is the set of points that are either in Span$\{\mbf{u}\}$ or in
Span$\{\mbf{v}\}$.
}

\item T / F: \qquad
If an augmented matrix has a row $[0 \; 0\; 0\; 7\; 0]$ then it
cannot be consistent.

\item T / F: \qquad
It can happen that the span of three vectors in $\RR^3$ is a line.

\item T / F: \qquad
\parbox{4in}{
By using different sequences of elementary
row operations, a matrix can be transformed into
two different reduced echelon forms.
}

%The linear system with augmented matrix has the same solution set
%The solution set of a linear system
%cannot change when elementary row operations are per

\item T / F: \qquad
The matrix $\left[\begin{array}{rrrr}0&1&0&1\\0&0&0&1\end{array}\right]$
is in reduced echelon form.

\een

\een

\end{document}
