%  bill.tex
%  Uncle Bill the billionaire
%  Linear algebra project


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\centerline{\bigrm Uncle Bill the Billionaire}
\bigskip

The purpose of this project is to demonstrate
how concepts from linear algebra
can be applied.  Follow the steps below.

An eccentric billionaire Bill tells his niece Nancy that
he wishes to start distributing some of his wealth to his relatives
before he dies in order to avoid inheritance taxes.  Bill gives Nancy a
choice as to how she may receive her money.  The first choice is for
Nancy to receive no money the first week, one cent the second week, and
each week after the first two
she is to receive five times the amount she received the week
before minus 6 times what she received two weeks before.  Her second
choice is to receive no money the first week, one dollar the second week,
and each week thereafter to receive the sum of the amounts she received
the previous two weeks.  Bill plans to make the same offer to all ten
of his
nephews and nieces.  He will continue paying each one according
to the plan each picks until he exhausts a supply of a billion
dollars he set aside.

Nancy knows you are currently taking linear algebra.  She therefore
naturally asks you for your advice.  Of course, Nancy wishes to get as
much of the billion as she can. (She never did like her cousins and she
has no sibling.) Follow the steps below to try to
determine which choice she should take.

\item{1.}Let $x_n$ and $y_n$ be the amounts Nancy is to be given in the
         $n^{\rm th}$ week assuming the two choices.  Write the formulas
         which determine the sequences $x_1,x_2,\cdots$ and
         $y_1,y_2,\cdots$
\item{2.}Consider first Nancy's first option.
         Instead of paying no money the first week and one cent the
         second week, Bill could have started by paying any amount the
         first week and any other amount the second week and then continued
         according to his plan.  For each choice of amounts
         paid the first two weeks one would obtain a sequence.  Show
         that the set of all such sequences forms a vector space.
\item{3.}Find the dimension of the vector space in Part~2 and prove you
         are right.
\item{4.}Find all vectors in part~2 that are of the form $x_n=r^n$.  Can
         you find a basis of vectors of this form.  If so, prove it.  If
         not, find a basis using as many vectors of the form $x_n=r^n$
         as you can.
\item{5.}Based on your results, determine a formula for how much money
         Nancy will receive each week if she chooses the first option.
\item{6.}Do the same for the second option.  You need not do the
         corresponding proofs in this case, just do the calculations.
\item{7.}Which choice will be better for Nancy?

\bye
