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%Tutorial slides on Python.

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% Author: FOSSEE 

% Copyright (c) 2009, FOSSEE, IIT Bombay

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% Title page

\title[Matrices \& Equations]{Python for Science and Engg: Matrices \& Solution of equations}

\author[FOSSEE] {FOSSEE}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

\date[] {31, October 2009\\Day 1, Session 4}

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%% the beginning of each subsection:

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    \frametitle{Outline}

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% DOCUMENT STARTS

\begin{document}

\begin{frame}

  \titlepage

\end{frame}

\begin{frame}

  \frametitle{Outline}

  \tableofcontents

%  \pausesections

\end{frame}

\section{Matrices}

\begin{frame}

\frametitle{Matrices: Introduction}

\end{frame}

\subsection{Initializing}

\begin{frame}[fragile]

\frametitle{Matrices: Initializing}

\begin{lstlisting}

In []: A

Out[]: 

\end{lstlisting}

\end{frame}

\subsection{Basic Operations}

\begin{frame}[fragile]

\frametitle{Transpose of a Matrix}

\begin{lstlisting}

Out[]:

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Sum of all elements}

  \begin{lstlisting}

Out[]: 12

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Matrix Addition}

  \begin{lstlisting}

Out[]: 

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Matrix Multiplication}

\begin{lstlisting}

Out[]: 

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Inverse of a Matrix}

\begin{small}

\begin{lstlisting}

Out[]: 

\end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Determinant}

\begin{lstlisting}

In []: det(A)

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\begin{small}

\begin{lstlisting}

Out[]: 

(array([-1.,  8., -1.]),

        [ 0.2981424 ,  0.33333333, -0.78664085],

        [ 0.59628479,  0.66666667,  0.58643303]]))

Out[]: array([-1.,  8., -1.])

\end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Computing Norms}

\begin{lstlisting}

Out[]: 8.1240384046359608

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \begin{small}

  \begin{lstlisting}

[[ -6.66666667e-01,  -1.23702565e-16,   7.45355992e-01],

 [ -3.33333333e-01,  -8.94427191e-01,  -2.98142397e-01],

 [ -6.66666667e-01,   4.47213595e-01,  -5.96284794e-01]]),

 array([ 8.,  1.,  1.]),

        [-0.        ,  0.89442719, -0.4472136 ],

        [-0.74535599,  0.2981424 ,  0.59628479]]))

  \end{lstlisting}

  \end{small}

\inctime{15}

\end{frame}

\section{Solving linear equations}

\begin{frame}[fragile]

\frametitle{Solution of equations}

Consider,

  \begin{align*}

    3x + 2y - z  & = 1 \\

    2x - 2y + 4z  & = -2 \\

    -x + \frac{1}{2}y -z & = 0

  \end{align*}

Solution:

  \begin{align*}

    x & = 1 \\

    y & = -2 \\

    z & = -2

  \end{align*}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving using Matrices}

Let us now look at how to solve this using \kwrd{matrices}

  \begin{lstlisting}

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution:}

\begin{lstlisting}

In []: x

Out[]: 

array([[ 1.],

       [-2.],

       [-2.]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Let's check!}

\begin{lstlisting}

In []: Ax

Out[]: 

\end{lstlisting}

\begin{block}{}

The last term in the matrix is actually \alert{0}!\\

We can use \kwrd{allclose()} to check.

\end{block}

\begin{lstlisting}

In []: allclose(Ax, b)

Out[]: True

\end{lstlisting}

\inctime{15}

\end{frame}

\subsection{Exercises}

\begin{frame}[fragile]

Solve the set of equations:

\begin{align*}

  x + y + 2z -w & = 3\\

  2x + 5y - z - 9w & = -3\\

  2x + y -z + 3w & = -11 \\

  x - 3y + 2z + 7w & = -5\\

\end{align*}

\inctime{10}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution}

\begin{align*}

  x & = -5\\

  y & = 2\\

  z & = 3\\

  w & = 0\\

\end{align*}

\end{frame}

\section{Summary}

\begin{frame}

  \begin{itemize}

  \item Matrices

    \begin{itemize}

      \item Transpose

      \item Addition

      \item Multiplication

      \item Inverse of a matrix

      \item Determinant

      \item Norms

    \end{itemize}

  \item Solving linear equations

  \end{itemize}

\end{frame}

\end{document}