\documentclass[letterpaper,11pt]{article}
\title{18.100B Lecture Notes}
\renewcommand{\today}{February 14, 2006}

\input{../share/100B.tex}

\begin{document}
\maketitle

\definition{\dt{$\C$} is the set $\R^2$ containing pairs $(a,b)$ with
  operations:
  \begin{description}
  \item[$(a,b)+(c,d)$] = $(a+c,b+d)$
  \item[$(a,b)*(c,d)$] = $(ac-bd,ad+bc)$
  \end{description}
  We write $(a,b)$ as $a+bi$, with $i = (0,1)$.

Note $\C$ contains $\R$ as a subfield in the form $(a,0)$. We will
write $\Re(a+ib) = a$, and $\Im(a+ib) = b$
}

\definition{The \dt{complex conjugate} of $z=a+ib$ is
  $\overline{z}=a-ib$

The complex conjugate has the following properties:
\alphlist {
  \item $\conj{(z+w)} = \conj{z} + \conj{w}$
  \item $\conj{(zw)} = \conj{z}\conj{w}$
  \item $z + \conj{z} = 2\Re(z)$ and $z - \conj{z} = 2\Im(z)$
  \item $z\conj{z}$ is real and positive (if $z \ne 0$)
}
}

\definition{$|z| = \sqrt{z\conj{z}}$}

\claim{
  \mbox{}
\alphlist {
  \item $|z|>0$ if $z\ne 0$, and $0$ otherwise
  \item $|z| = |\conj{z}|$
  \item $|zw| = |z||w|$
  \item $|\Re z| \leq |z|$
  \item $|z+w| \leq |z| + |w|$ (``Triangle inequality'')
}
}

\definition{The vector space over $\C$($C^n$) consists of $n$-tuples
  $\vec{z} = (z_1, z_2, ... z_n)$ $z_n \in \C$
  \begin{description}
  \item[$\vec{z} + \vec{w}$] = $(z_1+w_z, z_2+w_2, ... z_n+w_n)$
  \item[if $\lambda \in \Z, \lambda\vec{z}$] = $(\lambda z_1, \lambda
    z_2, ... \lambda z_n)$
  \end{description}
}
\definition{The \dt{Hermition inner product} is
  $$<\vec{z},\vec{w}> = \sum_{k=1}^{n}z_k\conj{w_k}$$
  Note $<\vec{z},\vec{z}> = \sum_{k=1}^{n}z_k\conj{z_k} \in \R, >0$ $\vec{z}\ne(0,0,...)$
}
\definition{$||\vec{z}|| = \sqrt{<\vec{z},\vec{z}}$}

\claim{
  \alphlist{
    \item $<\vz,\vw> = \conj(<\vw,\vz>)$
    \item $<\lambda\vz,\vw> = \lambda<\vz,\vw>$ \\
          $<\vz,\lambda\vw> = \conj{\lambda}<\vz,\vw>$
    \item $<\vz+\vw, \vec{s}> = <\vz,\vec{s}> + <\vw, \vec{s}>$
  }
}

\theorem{(Cauchy-Schwartz inequality)
  $$|<\vz,\vw>| \leq ||\vz||\cdot||\vw||$$
  Equivalently:
  $$|<\vz,\vw>|^2 \leq <\vz,\vz><\vw,\vw>$$
  $$\sum_{k=1}^nz_k\conj{w_k} \leq (\sum_{k=1}^nz_k\conj{z_k})(\sum_{k=1}^nw_k\conj{w_k})$$
}
\proof{
  \begin{eqnarray*}
    0 & \leq & <\vz - \lambda\vw,\vz - \lambda\vw> \\
      & =    & <\vz,\vz> - \lambda<\vw,\vz> \\ && - \conj{\lambda}<\vz,\vw>
    - \lambda\conj{\lambda}<\vw,\vw> \\
    & \mbox{Set $\lambda = \frac{<\vz,\vw>}{<\vw,\vw>}$} \\
    0 & \leq & <\vz,\vz> - \frac{<\vz,\vw>}{<\vw,\vw>}<\vw,\vz> \\
    && - \frac{<\vw,\vz>}{<\vw,\vw>}<\vz,\vw> +
    \frac{<\vz,\vw><\vw,\vz>}{<\vw,\vw>} \\
    <\vz,\vw><\vw,\vz> & \leq & <\vz,\vz><\vw,\vw> \\
    |<\vz,\vw>^2| & \leq & <\vz,\vz><\vw,\vw>
  \end{eqnarray*}
}

\end{document}