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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2270 Extra Credit Problems \\
\bf Chapter 1 \\
{\small\bf December 2011}
\end{center}

These problems were created for Bretscher's textbook, but apply for Strang's book, except for the division by chapter.
To find the background for a problem, consult Bretscher's textbook, which can be checked out from the math library or the LCB Math Center.

{\bf Due date}: See the internet due dates.
Records are locked on that date and only corrected,
never appended.

{\bf Submitted work}. Please submit one stapled package.
Kindly label problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number. You may attach this printed sheet to
simplify your work.

\Prob{XC1.1-26}{(Three possibilities)}
Determine which values of $k$ correspond to (a) a unique solution, (b)
no solution or (c) infinitely many solutions.
%3x3
$$
\begin{array}{|rcrcrcl|}
x &+& 2y &+& kz &=& 0 \\
2x &+& 4y &+& (k+1)z &=& 2 \\
3x &+& 6y &+& (2k+1)z &=& 2 \\
\end{array}
$$

\Prob{XC1.1-30}{(Polynomial interpolation)}
Find the polynomial $f(x)=a+bx+cx^2$ which passes through the points
$(1,9)$, $(2,24)$, $(3,47)$.
% f:=x->2+3*x+4*x^2;

\Prob{XC1.1-32}{(Polynomial interpolation)}
Find all polynomials $f(x)=a+bx+cx^2$ which pass through the points
$(1,10)$, $(2,28)$ and $f'(3)=33$.
% f:=x->2+3*x+5*x^2;

\Prob{XC1.2-22}{(RREF)}
Report five types of $3\times 4$ matrices in RREF form.

\Prob{XC1.2-28}{(Combo rule)}
Consider the following systems.

\begin{equation}\label{2-28.eq1}
\begin{array}{|lccclcl|}
a_{11}x_1 &+& \cdots &+& a_{1n}x_n &=& b_1 \\
          & & \vdots & &           & &     \\
a_{m1}x_1 &+& \cdots &+& a_{mn}x_n &=& b_m
\end{array}
\end{equation}

\medskip

\begin{equation}\label{2-28.eq2}
\begin{array}{|lccclcl|}
a_{11}x_1 &+& \cdots &+& a_{1n}x_n &=& b_1 \\
          & & \vdots & &           & &     \\
c_{k1}x_1 &+& \cdots &+& c_{kn}x_n &=& d_k  \\
          & & \vdots & &           & &     \\
a_{m1}x_1 &+& \cdots &+& a_{mn}x_n &=& b_m
\end{array}
\end{equation}
We assume the systems identical except for equation $k$, which is obtained
in the second system by applying a combination rule {\tt combo(r,k,c)}
to the first system. We assume $r \ne k$ and then the coefficients in the second system are given by
$$ c_{kj}=a_{kj}+ca_{rj},\quad j=1,\ldots,n,\quad d_k=b_k+cb_r. $$
(a) Prove that every solution of system \rf{2-28.eq1} is a solution of system \rf{2-28.eq2}.

(b) Prove that every solution of system \rf{2-28.eq2} is a solution of system \rf{2-28.eq1}.


\Prob{XC1.2-30}{(Polynomial interpolation)}
Find the polynomial $f(x)=a+bx+cx^2+dx^3$ which satisfies $f(1)=8$,
$f(2)=24$, $f(3)=24$, $f(4)=110$.

\Prob{XC1.3-26}{(Matrix algebra)}
Find a $3\times 3$ matrix $A$ which satisfies the following relations.
$$
A\colvector{1\\ 1 \\ 1} = \colvector{1 \\ 0 \\ 1}, \quad
A\colvector{1\\ 0 \\ 1} = \colvector{0 \\ 0 \\ 1}, \quad
A\colvector{0\\ 0 \\ 1} = \colvector{1 \\ 1 \\ 0}
$$

{\bf End of extra credit problems chapter 1.}

\end{document}
