added all my stuff

linearalgebra/exercises/correlations.tex

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+\documentclass[addpoints,10pt]{exam}
+\usepackage{url}
+\usepackage{color}
+\usepackage{hyperref}
+\usepackage{graphicx}
+\usepackage{amsmath}
+
+\pagestyle{headandfoot}
+\runningheadrule
+\firstpageheadrule
+
+\firstpageheader{Scientific Computing}{Principal Component Analysis}{Oct 29, 2014}
+%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
+\firstpagefooter{}{}{}
+\runningfooter{}{}{}
+\pointsinmargin
+\bracketedpoints
+
+%\printanswers
+\shadedsolutions
+
+\usepackage[mediumspace,mediumqspace,Gray]{SIunits}      % \ohm, \micro
+
+%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{listings}
+\lstset{
+ basicstyle=\ttfamily,
+ numbers=left,
+ showstringspaces=false,
+ language=Matlab,
+ breaklines=true,
+ breakautoindent=true,
+ columns=flexible,
+ frame=single,
+ captionpos=t,
+ xleftmargin=2em,
+ xrightmargin=1em,
+ aboveskip=10pt,
+ %title=\lstname,
+ title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
+ }
+
+
+\begin{document}
+
+\sffamily
+%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{questions}
+  \question \textbf{Gaussian distribution}
+  \begin{parts}
+    \part Use \texttt{randn} to generate 1000000 normally (zero mean, unit variance) distributed random numbers.
+    \part Plot a properly normalized histogram of these random numbers.
+    \part Compare the histogram with the probability density of the Gaussian distribution
+    \[ p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
+    where $\mu$ is the mean and $\sigma^2$ is the variance of the Gaussian distribution.
+    \part Generate Gaussian distributed random numbers with mean $\mu=2$ and
+    standard deviation $\sigma=\frac{1}{2}$.
+  \end{parts}
+
+  \question \textbf{Covariance and correlation coefficient}
+  \begin{parts}
+    \part Generate two vectors $x$ and $z$ with Gausian distributed random numbers.
+    \part Compute $y$ as a linear combination of $x$ and $z$ according to
+    \[ y = r \cdot x + \sqrt{1-r^2}\cdot z \]
+    where $r$ is a parameter $-1 \le r \le 1$.
+    What does $r$ do?
+    \part Plot a scatter plot of $y$ versus $x$ for about 10 different values of $r$.
+    What do you observe?
+    \part Also compute the covariance matrix and the correlation
+    coefficient matrix between $x$ and $y$ (functions \texttt{cov} and
+    \texttt{corrcoef}). How do these matrices look like for different
+    values of $r$? How do the values of the matrices change if you generate
+    $x$ and $z$ with larger variances?
+    \part Do the same analysis (Scatter plot, covariance, and correlation coefficient)
+    for \[ y = x^2 + 0.5 \cdot z \]
+    Are $x$ and $y$ really independent?
+  \end{parts}
+
+  \question \textbf{Principal component analysis}
+  \begin{parts}
+    \part Generate pairs $(x,y)$ of Gaussian distributed random numbers such
+    that all $x$ values have zero mean, half of the $y$ values have mean $+d$
+    and the other half mean $-d$, with $d \ge0$.
+    \part Plot scatter plots of the pairs $(x,y)$ for $d=0$, 1, 2, 3, 4 and 5.
+    Also plot a histogram of the $x$ values.
+    \part Apply PCA on the data and plot a histogram of the data projected onto
+    the PCA axis with the largest eigenvalue.
+    What do you observe?
+  \end{parts}
+  
+\end{questions}
+
+
+\end{document}

linearalgebra/exercises/matrices.tex

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+\documentclass[addpoints,10pt]{exam}
+\usepackage{url}
+\usepackage{color}
+\usepackage{hyperref}
+\usepackage{graphicx}
+\usepackage{amsmath}
+
+\pagestyle{headandfoot}
+\runningheadrule
+\firstpageheadrule
+
+\firstpageheader{Scientific Computing}{Matrix multiplication}{Oct 28, 2014}
+%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
+\firstpagefooter{}{}{}
+\runningfooter{}{}{}
+\pointsinmargin
+\bracketedpoints
+
+%\printanswers
+\shadedsolutions
+
+\usepackage[mediumspace,mediumqspace,Gray]{SIunits}      % \ohm, \micro
+
+%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{listings}
+\lstset{
+ basicstyle=\ttfamily,
+ numbers=left,
+ showstringspaces=false,
+ language=Matlab,
+ breaklines=true,
+ breakautoindent=true,
+ columns=flexible,
+ frame=single,
+ captionpos=t,
+ xleftmargin=2em,
+ xrightmargin=1em,
+ aboveskip=10pt,
+ %title=\lstname,
+ title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
+ }
+
+
+\begin{document}
+
+\sffamily
+%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{questions}
+  \question \textbf{Matrix multiplication} 
+  Calculate the results of the following matrix multiplications and
+  confirm the result using matlab.
+    \[ \begin{pmatrix} 2 \\ -4 \\ -1 \end{pmatrix} \cdot
+    \begin{pmatrix} 3 & -4 & -4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 3 & -3 & -1 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 4 & -1 & 2 \\ -1 & 3 & 1 \\ 4 & -2 & 1 \\ 4 & -3 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} -2 & -2 & 0 & -3 \\ 3 & -2 & 1 & 0 \\ 1 & -2 & -4 & 0 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 3 & 1 \\ 1 & 4 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & -3 & 4 & 1 \\ -2 & -1 & -2 & -3 \\ -3 & 1 & -2 & -3 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 1 & 1 & -4 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 \\ 2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 3 & 1 & -2 \\ 2 & 1 & 3 \\ 1 & 1 & 2 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 & 2 \\ -3 & 3 \\ -4 & 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 3 \\ 2 \end{pmatrix} \cdot
+    \begin{pmatrix} -3 & 2 & -4 & 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -1 \\ -4 \\ -1 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & -4 & 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 4 & -2 & -2 & -4 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 \\ 2 \\ 1 \\ -1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -2 & -3 & -4 \\ 1 & 3 & 2 \\ -4 & -2 & 1 \end{pmatrix} \cdot
+    \begin{pmatrix} 1 & 2 & -2 & 4 \\ 3 & -1 & 1 & -1 \\ -3 & 2 & -1 & 2 \end{pmatrix} = \]
+
+   \[ \begin{pmatrix} 2 & -4 & 4 & 4 \\ -3 & 3 & 2 & 1 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & 3 & 4 & -2 \\ -4 & -2 & -1 & 0 \\ 1 & 2 & -4 & -4 \\ 3 & 2 & -2 & -4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 3 & 1 & -2 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} -4 \\ 3 \\ -2 \\ 4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -1 & 3 & 4 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 \\ 4 \\ -3 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 1 & -4 & 3 & 3 \end{pmatrix} \cdot
+    \begin{pmatrix} 1 \\ 0 \\ -4 \\ -1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -4 & -4 & -3 \\ -2 & -2 & 4 \\ -3 & 4 & -3 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & 3 & -4 & 4 \\ -1 & -2 & -3 & 1 \\ 1 & -2 & 2 & 0 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -3 & 0 & 4 & 1 \\ 0 & 1 & 1 & 4 \end{pmatrix} \cdot
+    \begin{pmatrix} -4 & 3 & 1 & 4 \\ 1 & -4 & 1 & -3 \\ -4 & 0 & -4 & -4 \\ 1 & -2 & 4 & 4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 4 \\ 3 \\ 4 \\ -2 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 & 4 & 3 & 3 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 1 & 2 & 0 & 3 \end{pmatrix} \cdot
+    \begin{pmatrix} -3 \\ 1 \\ 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -4 & 0 & -1 & 3 \\ 0 & -4 & 3 & -3 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 & -4 & -1 \\ 3 & 2 & 0 \\ -2 & 3 & -2 \\ 1 & 2 & -2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 2 & 0 & 3 \\ 1 & -4 & -1 \\ 3 & 0 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & 2 & -1 & -2 \\ -1 & -1 & -3 & 4 \\ 2 & 4 & -4 & 1 \end{pmatrix} = \]
+    \[ \begin{pmatrix} -1 & 4 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -4 & 3 \\ -4 & 0 \\ -2 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & 1 & -4 & 2 \\ 2 & 3 & -2 & -1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -2 & -1 \end{pmatrix} \cdot
+    \begin{pmatrix} 1 \\ -2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -2 & 2 & -2 & -3 \\ 2 & -4 & -2 & 2 \\ 0 & 2 & -2 & -2 \\ 1 & -2 & -2 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} 1 & -2 & 2 \\ -4 & -2 & -2 \\ 3 & 1 & 4 \\ -4 & 1 & -2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -1 & -3 & 0 & -1 \\ 4 & -2 & 1 & 2 \end{pmatrix} \cdot
+    \begin{pmatrix} -3 & -4 \\ -4 & 0 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -1 & 1 & -2 \\ -2 & 2 & -4 \\ 1 & -2 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 & 2 & -4 \\ 1 & 3 & 0 \\ 1 & 4 & -4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -3 & 3 \\ -3 & 2 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 & -3 \\ -2 & -4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 1 & 1 & -3 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 \\ -2 \\ 3 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -4 & 2 & 1 \\ 4 & 0 & -2 \\ 2 & 3 & -3 \\ -2 & -2 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 & 2 & 0 & -2 \\ 2 & -2 & 0 & -1 \\ -4 & 3 & -3 & 4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -2 & -4 & 2 & 4 \\ 3 & -3 & 2 & 1 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & 4 & -1 & -4 \\ 2 & 3 & -4 & -1 \\ 3 & 2 & -2 & 4 \end{pmatrix} = \]
+
+   \[ \begin{pmatrix} -3 & -2 & -1 & -3 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 \\ -2 \\ 3 \\ -2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 4 & 4 & 2 & 3 \end{pmatrix} \cdot
+    \begin{pmatrix} 3 \\ 3 \\ -2 \\ 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 3 & 2 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 \\ 4 \\ 3 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 2 & -1 & 0 & -2 \\ 0 & -4 & -3 & -1 \end{pmatrix} \cdot
+    \begin{pmatrix} 4 & -3 & 2 & 4 \\ -3 & -4 & 1 & 1 \\ 1 & 3 & -2 & 3 \\ -1 & -2 & 3 & 0 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -3 & -3 & 3 & 2 \\ 2 & 2 & -3 & 1 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & 1 \\ 4 & 2 \\ -3 & -1 \\ -3 & 4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -4 & -3 \end{pmatrix} \cdot
+    \begin{pmatrix} -2 \\ 3 \\ 4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 4 & 4 \end{pmatrix} \cdot
+    \begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 1 & -2 & 3 \end{pmatrix} \cdot
+    \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -3 & 2 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 \\ 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -2 & -4 & -4 & 0 \\ 0 & 3 & 4 & -4 \\ 4 & 2 & -2 & -4 \\ 0 & 0 & 4 & -1 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & -1 \\ -1 & 1 \\ -4 & -3 \\ 2 & 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -3 \\ 3 \\ -3 \\ -4 \end{pmatrix} \cdot
+    \begin{pmatrix} 2 & 4 & -2 & 1 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 2 \\ 0 \end{pmatrix} \cdot
+    \begin{pmatrix} -1 & -3 & -2 & 2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 0 & -4 & -4 & 4 \end{pmatrix} \cdot
+    \begin{pmatrix} 1 \\ 4 \\ 0 \\ 4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -3 & -1 \\ -3 & -1 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & -3 & 3 & -2 \\ -4 & 1 & -1 & 4 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 4 & 0 \\ -1 & 4 \\ 1 & -3 \end{pmatrix} \cdot
+    \begin{pmatrix} -4 & -4 \\ -4 & 2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -1 \\ 3 \\ 2 \\ 4 \end{pmatrix} \cdot
+    \begin{pmatrix} 0 & -1 & 0 & 0 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 3 \\ -2 \\ 2 \\ 3 \end{pmatrix} \cdot
+    \begin{pmatrix} -2 & -3 & -4 & 2 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} 2 & -2 & -4 & 4 \\ 0 & 1 & -3 & -2 \\ -1 & 3 & 0 & -2 \end{pmatrix} \cdot
+    \begin{pmatrix} -4 & 1 \\ -4 & 3 \end{pmatrix} = \]
+
+    \[ \begin{pmatrix} -4 & -1 & 3 \end{pmatrix} \cdot
+    \begin{pmatrix} -4 \\ -3 \\ 3 \end{pmatrix} = \]
+
+  \question \textbf{Automatic generation of exercises}
+  Write some matlab code that generates exercises like this one automatically! :-) 
+  
+\end{questions}
+
+
+\end{document}