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\begin{document}
\centerline{Math 201 Homework for Tuesday, Week 10}
\bigskip
Due: Tuesday, November 9.

\begin{prob}
  Compute the determinants of the following matrices by using the permutation expansion.
  \begin{enumerate}
   \item $\begin{pmatrix} -1 & 2+i & 3\\ 1-i & i & 1 \\ 3i & 2 & -1+i \end{pmatrix}$ \qquad \qquad 
   (b) $\begin{pmatrix} 0& 1 & 0 & 0 &  0\\ 0& 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 4 & 0\\ 5 & 0 & 0 & 0 & 0 \end{pmatrix}$
  \end{enumerate}
\end{prob}

\begin{prob}
  Compute the determinants of the same matrices as in Problem 1 by using the Laplace expansion along any row or column. Be clear about which row or column you are using.
\end{prob}
\begin{prob}
    Let $p(x_1,\dots,x_n)$ be a polynomial on $n$ variables with coefficients in a field $F$. An arbitrary term of this polynomial is of the form $ax_1^{d_1}x_2^{d_2}\dots x_n^{d_n}$, where $a\in F$ and $d_i$ is a nonnegative integer for all $i$. The total degree of this term is $d_1+\dots +d_n$.  
  
    For example, the polynomial
    \[p(x_1,x_2,x_3)=2+x_1+3x_1x_2^2-4x_2^2x_3^3+9x_1x_2x_3\]
    has five terms of total degree 0, 1, 3, 5, and 3, respectively. 
  
  
    Here is a result from polynomial algebra. If $p$ satisfies the conditions:
    \begin{itemize}
     \item[(i)] $p(x_1,\dots, x_n)=0$ whenever $x_i=x_j$ for $i\neq j$;
     \item[(ii)] the total degree of every term is $n(n-1)/2$,
    \end{itemize}
    then 
    \[p(x_1,\dots,x_n)=k(x_2-x_1)(x_3-x_1)\cdots(x_n-x_{n-1})\]
     for some $k\in F$. Here the product contains all the terms of the form $x_j-x_i$ with $1\leq i < j \leq n$. Note that the coefficient $k$ is equal to the coefficient of $x_2x_3^2\cdots x_{n-1}^{n-2}x_n^{n-1}$ in $p$.
  
    For example, when $n=2$ and $F=\R$, $p(x_1,x_2)=x_1-x_2$ satisfies both properties, $p(x_1,x_2)=x_1^2-x_2^2$ satisfies (i) but not (ii), and $p(x_1,x_2)=x_1+2x_2$ satisfies (ii) but not (i).
  
    Now consider the Vandermonde matrix
    \[V(x_1,\dots,x_n)= \begin{pmatrix} 1 & 1 & \cdots & 1\\
    x_1 & x_2 & \cdots & x_n \\
    x_1^2 & x_2^2 & \cdots & x_n^2\\
    \vdots & \vdots & \ddots & \vdots \\
     x_1^{n-1} & x_2^{n-1} & \cdots & x_n^{n-1}
     \end{pmatrix}\]
     Let $p(x_1,\dots,x_n)=\det(V(x_1,\dots,x_n))$. 
      \begin{enumerate}
      \item Using properties of determinants, prove that $p$ satisfies property (i).
      \item Using the permutation expansion of the determinant, prove that $p$ satisfies property (ii). 
  
      (\emph{Hint:} As always, it is useful to play around with small cases of $n$ to understand what is really going on. Try $n=2$ and $n=3$, and then use what you learn to argue for the general case. When you write your solutions, you should write them for arbitrary $n$.)
      \item It follows from (a) and (b) and the discussion above that 
  
      \[p(x_1,\dots,x_n)=k(x_2-x_1)(x_3-x_1)\cdots(x_n-x_{n-1})\]
       for some $k\in F$. Find the value of the coefficient $k$.
      \item (BONUS.) A general polynomial of degree~$d$ in one
        variable over the real numbers has the form
        \[
  	q(x)=a_0+a_1x+a_2x^2+\dots+a_d x^d,
        \]
        where the~$a_i$ are real numbers.  Pick~$n$ distinct real
        numbers~$x_1,\dots,x_n$, and pick arbitrary (not necessarily distinct)
        real numbers~$b_1,\dots,b_n$.  Prove that there is a unique
        polynomial~$q(x)$ of degree~$n-1$ over the real numbers such that~$q(x_i)=b_i$
        for~$i=1,\dots,n$.
      \item (BONUS.)  Use the Vandermonde determinant to prove
        that the collection of functions~$\left\{ e^{\alpha x}: \alpha\in\R
        \right\}$ is linearly independent.  (Recall that the set of functions
        from~$\R$ to~$\R$ is a vector space.  The solution to this problem would
        show that this space is infinite dimensional.)
      \end{enumerate}
\end{prob}
\end{document}
