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Math 192r, Problem Set \#5 \\
(due 10/4/01)
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\begin{enumerate}
\item
There is a unique polynomial of degree $d$ such that $f(k)=2^k$ for $k=0,1,...,d$. What is $f(d+1)$? What is $f(-1)$?
\item
One basis for the space of polynomials of degree less than $d$
is the monomial basis $1,t,t^2,...,t^{d-1}$.
Another is the shifted monomial basis $1,(t+1),(t+1)^2,...,(t+1)^{d-1}$.
Call these bases $u_1,...,u_d$ and $v_1,...,v_d$ respectively.
\begin{itemize}
\item[(a)]
Derive a formula for the entries of
the change-of-basis matrix $M$ expressing the $u_i$'s
as linear combinations of the $v_j$'s.
\item[(b)]
Derive a formula for the entries of
the change-of-basis matrix $N$ expressing the $v_j$'s
as linear combinations of the $u_i$'s.
\item[(c)]
From the description of $M$ and $N$ as basis-change matrices,
we know that $MN = NM = I$.
Forgetting for the moment what $M$ and $N$ mean,
rewrite the assertions $MN = NM = I$
as binomial coefficient identities,
and prove them either algebraically or bijectively.
\end{itemize}
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