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\pagestyle{myheadings} \markright{Math 4310 (Fall 2020) \hspace{2in}
(HW9)}

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  {\large Homework \# 9

    \medskip
    Math 4310 Fall 2020

    \medskip
    
    Due Thursday evening, 11/12/20 (at midnight).
  }

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Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

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\noindent {\bf Journal entry.} There is no journal entry this week.

\vskip 0.1in

\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}

\item
  Prove the Cauchy-Schwarz formula: if $V$ is an inner product space (not necessarily of finite
  dimension), and ${\bf v}$ and ${\bf w}$ are elements of $V$, then
  \[ | \langle {\bf v}, {\bf w} \rangle | \le ||{\bf v}||\, ||{\bf w}||. \]
  ({\it Hint:} Consider the projection of ${\bf w}$ onto the span of ${\bf v}$.  Note that the
  length of the projection is no more than the length of ${\bf w}$ (why?)).

  \medskip
  Prove the following statements.
  \babc
\item Prove that for all positive real numbers $a,b,c,d$, 
  \[ 16 \le (a + b + c + d) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \right). \]
\item Prove that for all positive integers $n$ and all real numbers $a_1, \ldots, a_n$, then
  \[ (a_1 + \ldots + a_n)^2 \le n (a_1^2 + \ldots + a_n^2). \]
  (equivalently, the square of the average of a set of numbers is no more than the average of the
  squares of the numbers).
\item Prove that for continuous functions $f$ and $g$ on the interval $[0, 1]$, that
  \[
  \left| \int_0^1 f(x) g(x) dx \right|^2 \le \left| \int_0^1 f(x)^2 dx \right| \, \left| \int_0^1 g(x)^2 dx \right|
  \]
  \eabc

\pagebreak  
  \item In a triangle with sides of lengths $a$, $b$, and $c$,
    let $d$ be the length of the line segment from the midpoint of the
    side with length $c$ to the oppositve vertex. Show that $a^2 + b^2 = \frac{1}{2} c^2 + 2 d^2$.

    \includegraphics[width=60mm]{appolonius.png}
        
\item Apply the Gram-Schmidt process to find an orthonormal basis of
  the space $\R[x]_{\le 2}$ of polynomials of degree at most 2, where the inner product
  is given by
  \[ \langle f, g \rangle = \int^1_0 f(x) g(x) \, dx. \]


\item Let $V = \R^n$.
  As usual, we think of each ${\bf v}$ in $\R^n$ as a column vector, that is, as a
  $n \times 1$ matrix.  We also equate scalars and $1 \times 1$ matrices.
  \babc
\item Suppose that $A$ is a symmetric $n \times n$ matrix, and we define
  \[ \langle {\bf v}, {\bf w} \rangle = {\bf v}^T A {\bf w}, \]
  show that this is an inner product if and only if $A$ satisfies: for ${\bf x} \ne {\bf 0} \in \R^n$
  then ${\bf x}^T A {\bf x} > 0$ (such a symmetric matrix is called
  {\bf positive definite}).
\item Suppose that $\langle \cdot, \cdot \rangle$ is an inner product on $\R^n$.
  Show that there exists a positive definite matrix $A$ such that
  \[ \langle {\bf v}, {\bf w} \rangle = {\bf v}^T A {\bf w}, \]
\item If
  \[A = \begin{pmatrix} 6 & -2 \\ -3 & 1 \end{pmatrix},\]
  is the function defined by
  \[ \langle {\bf v}, {\bf w} \rangle = {\bf v}^T A {\bf w} \]
  (for all ${\bf v}, {\bf w} \in \R^2$) an inner product on $\R^2$?
    \eabc

  \item Suppose that $V$ is a inner product space over $\F = \R$ or $\C$, and suppose that
    $({\bf e}_1, \ldots, {\bf e}_n)$ is an orthonormal basis of $V$.  Suppose that
    ${\bf v}_1, \ldots, {\bf v}_n$ are vectors in $V$ such that
    \[ || {\bf e}_j - {\bf v}_j || < \frac{1}{\sqrt{n}} \]
    for each $1 \le j \le n$.  Prove that $({\bf v}_1, \ldots, {\bf v}_n)$ is a basis of $V$.
    
  \item For a subspace $W \subset V$ of a finite dimensional inner product space $V$ (over $\F = \R$ or $C$), the orthogonal complement is defined to be
    \[ W^\perp = \{ {\bf v} \in V \mid \langle {\bf v}, {\bf w} \rangle = 0,
    \mbox{ for all ${\bf w} \in W$} \} \]
    \babc
  \item Show that $V = W \oplus W^\perp$
  \item Show that, if $A$ is an $m \times n$ matrix, and $V = \R^n$ is equipped with the
    standard inner product, then
    \[ (\ker A)^\perp  = image(A^T) \]
    and conclude that
    \[ V = \ker A \oplus image(A^T). \]
    \eabc


\end{enumerate}

\end{document}



