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\header{{\bfseries\large Exercise 3}}{{\bfseries\large Matrices}}{{\bfseries\large 22. Oktober, 2019}}
\firstpagefooter{Dr. Jan Grewe}{Phone: 29 74588}{Email:
  jan.grewe@uni-tuebingen.de}
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\begin{document}
\vspace*{-6.5ex}
\begin{center}
  \textbf{\Large Introduction to Scientific Computing}\\[1ex]
  {\large Jan Grewe, Jan Benda}\\[-3ex]
  Neuroethology \hfill --- \hfill Institute for Neurobiology \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
\end{center}

The exercises are meant for self-monitoring and revision of the
lecture. You should try to solve them on your own. Your solution
should be submitted as a single script (m-file) in the Ilias
system. Each task should be solved in its own ``cell''. Each cell must
be executable on its own. The file should be named according to the
following pattern:
``matrices\_\{lastname\}.m''(e.g. matrices\_mueller.m).

\begin{questions}

  \question Create the following matrix:
  \[ A = \left( \begin{array}{ccc} 7 & 3 & 5 \\ 1 & 8 & 3 \\ 8 & 6 &
      4 \end{array} \right) \]
  \begin{parts}
    \part Use the function \code{size} to check for its size.
    \begin{solution}
      \code{x = [7 3 5; 1 8 3; 8 6 4];\\disp(size(x))}
    \end{solution}
    \part Use the help to figure out how to get only the size along a certain dimension. Display the sizes of each dimension.
    \begin{solution}
      \code{disp(size(x, 1))}\\\code{disp(size(x, 2))}
    \end{solution}
    \part Copy the content at the position 3rd line, 2nd column to a new variable.
    \begin{solution}
      \code{c = x(3, 2)}
    \end{solution}
    \part Display all elements of the  1st, 2nd and 3rd line.
    \begin{solution}
      \code{disp(x([1 2 3], :));}
    \end{solution}
    \part Display all elements of the 1st, 2nd, and 3rd column.
     \begin{solution}
      \code{disp(x(:, 1))\\ disp(x(:, 2))\\ disp(x(:, 3))}
    \end{solution}
    \part Increment all elements of the 2nd line and the 3rd column about 1 (reassign the result to the respective elements).
    \begin{solution}
      \code{x(2,:) = x(2,:) + 1;}\\
      \code{x(:,3) = x(:,3) + 1;}
    \end{solution}
    \part Subtract five from all elements of the 1st line.
    \begin{solution}
      \code{x(1,:) = x(1,:) - 5;}
    \end{solution}
    \part Multiply all elements of the 3rd column by 2.
    \begin{solution}
      \code{x(:,3) = x(:,3) .* 2;}
    \end{solution}
  \end{parts}

  \question Create a $5 \times 5$ matrix \code{M} that contains random numbers (use the function 
  \verb+randn()+. Use the help to find out what it does).
  \begin{parts}
    \part Display the content of \code{M} at position 2nd line and 3rd column.
    \begin{solution}
      \code{M = randn(5, 5);}
      \code{disp(M(2,3))}
    \end{solution}

    \part Display all elements of the 1st, 3rd and last line.
    \begin{solution}
      \code{disp(M(1,:)) \\ disp(M(3,:))\\ disp(M(size(M,1), :))}
    \end{solution}

    \part Display the elements of the 2nd and 4th column.
    \begin{solution}
      \code{disp(M(:,2))\\ disp(M(:,4))}
    \end{solution}
 
    \part Select with a single command all elements of every 2nd column and store them in a new variable.
    \begin{solution}
      \code{y = M(:, [2:2:size(M,2)])}
    \end{solution}

    \part Calculate the averages of lines 1, 3, and 5 (use the function \verb+mean+}, see help).
    \begin{solution}
      \code{mean(M([1 3 5],:), 2)}
    \end{solution}

    \part Calculate the sum of all elements in the 2nd and 4th column
    (function \code{sum}, see help).
    \begin{solution}
      \code{sum(M(:, [2 4]), 1)}
    \end{solution}

    \part Calculate the total sum of all elements in \code{M}
    \begin{solution}
      \code{sum(M(:))}
    \end{solution}

    \part Replace all elements of the 2nd line with those of the 4th line.
    \begin{solution}
      \code{M(2,:) = M(4,:)}
    \end{solution}

    \part Execute the following command: \code{M(1:2,1) = [1, 2,
      3]}. What could have been intended by the command and what does the error message tell?
    \begin{solution}
      \code{M(1:2,1) = [1, 2,3];\\ Subscripted assignment dimension
        mismatch.}  \\ Der einzuf\"ugende Vektor hat 3 Elemente, die
      Auswahl von M in die geschrieben werden soll hat nur die
      Gr\"o{\ss}e 2;
    \end{solution}
  \end{parts}

  \question Indexing in matrices can use the 
  \textit{subscript} indices or the \textit{linear} indices (you may want to check the help for the functions \verb+sub2ind+ and \verb+ind2sub+).
  \begin{parts}
    \part Create a 2-D matrix filled with random numbers and the size
    \verb+[10,10]+.
    \begin{solution}
      \code{x = randn(10, 10)}
    \end{solution}

    \part How many elements are stored in it?
    \begin{solution}
      \code{disp(numel(x))}
    \end{solution}

    \part Employ linear indexing to select 50 random values.
    \begin{solution}
      \code{x(randi(100, 50, 1)])}
    \end{solution}

    \part Can you imagine an advantage of using linear instead of subscript indexing?
    \begin{solution}
      Die Matrize ist 2-dimensional. Wenn mit dem subscript index
      zugegriffen werden soll, dann muss auf die Dimensionen
      einzeln geachtet werden. Mit dem linearen Indexieren kann einfach
      einen Vektor mit n Indices benutzt werden. Wenn es auch noch eine
      eindeutige (ohne doppelte) Auswahl sein soll, dann muss bei
      2-D viel komplexer kontrollieren.
    \end{solution}

    \part Calculate the total sum of all elements with a single command.
    \begin{solution}
      \code{sum(x(:))} or \code{sum(sum(x))}
    \end{solution}

  \end{parts}
  
  \question Create three variables \verb+x = [1 5 9]+ and
  \verb+y = [7 1 5]+ and \verb+M = [3 1 6; 5 2 7]+. Which of the
  following commands will pass? Which command will fail? If it fails,
  why? Test your predictions.
  \begin{parts}
    \part \code{x + y}
    \begin{solution}
      works!
    \end{solution}

    \part \code{x * M}
    \begin{solution}
      Matrixmultiplication will not work! Inner dimensions must agree!
    \end{solution}

    \part \code{x + y'}
    \begin{solution}
      Fail! Dimensionalities do not match.
    \end{solution}

    \part \code{M - [x y]}
    \begin{solution}
      Fail! \code{[x y] is a line vector of length 6, M is a martix.}
    \end{solution}

    \part \code{[x; y]}
    \begin{solution}
      Works! Size: 2  3
    \end{solution}

    \part \code{M - [x; y]}
    \begin{solution}
      Works!
    \end{solution}
  \end{parts}

  \question Create a 3-D matrix from two 2-D matrices. Use the
  function \code{cat} (check the help to learn its usage).
  \begin{parts}
    \part Select all elements of the first ``page'' (index 1 in the 3. dimension).
    \begin{solution}
      \code{x = randn(5,5); \\y = randn(5, 5);\\ z = cat(3, x, y);\\disp(z(:,:,1))}
    \end{solution}
  \end{parts}

  \question Create a $5 \times 5 \times 5$ matrix of random numbers that have been drawn from a uniform distribution. Values should be in the range 0 and 100.
  \begin{parts}
    \part Calculate the average of each ``page'' (function \verb+mean()+, see help).
    \begin{solution}
      \code{x = round(rand(5,5,5) .* 100);\\ disp(mean(mean(x(:,:,1))))\\ disp(mean(mean(x(:,:,2)))) \\ disp(mean(mean(x(:,:,3))))}
    \end{solution}
  \end{parts}
  \end{questions}

\end{document}