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\noindent
{\bf Artificial Neural Networks and Deep Learning} \hfill Summer 2018 \\
Christian Borgelt and Christoph Doell \hfill from 2018.04.24

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\vskip3ex
\centerline{\bf Organisational Rules for Exam Admission}

\medskip\noindent
The lecture is accompanied by exercise sheets.
At the beginning of each exercise lesson, the exercises are
{\em voted for}. By voting for an exercise, one expresses one's
willingness to present something about it. (Suggestions for a
solution will be discussed, they need not be correct right away.)
To the exam will be admitted who
\begin{enumerate}\itemsep0pt\parskip2pt
\item voted for {\em at least\/} half of the exercises {\em and}
\item presented something for {\em at least\/} two exercises.
\end{enumerate}

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\vskip3ex
\centerline{{\bf Exercise Sheet 1} \quad --- \quad
            Reminder: Vector Computations and Geometry}

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\subsubsection*{Exercise 1 \quad\rm Vector Computations}

Let the two vectors $\vec{x} = (3,1,2)^{\top}$ and
$\vec{y} = (2,3,4)^{\top}$ be given. Compute
\vspace*{-1mm}
\begin{enumerate}\itemsep0pt
\item[a)] the difference vector pointing from $\vec{x}$ to $\vec{y}$,
\item[b)] the scalar (or inner) product
          $\vec{x}^{\top}\! \cdot \vec{y}$,
\item[c)] the vector product $\vec{x} \times \vec{y}$.
\item[d)] the matrix (or outer) product $\vec{x} \cdot \vec{y}^{\top}$,
\item[e)] the angle between the two vectors.
\end{enumerate}

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\subsubsection*{Exercise 2 \quad\rm Hessian Normal Form}

Transform the following line equations into Hessian normal form
$\vec{r}^\top\! \cdot \vec{n}_0 - d = 0$:
\begin{enumerate}\itemsep0pt
\item[a)\hfill] $3x - 4y - 20 = 0$,
\item[b)\hfill] $x + y + 3 = 0$,
\item[c)\hfill] $y = b x + c$ mit $c < 0$.
\end{enumerate}

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\subsubsection*{Exercise 3 \quad\rm Distances to Straight Lines}

\begin{enumerate}\itemsep0pt
\item[a)\hfill] What distance does the origin have from the line
                $12x - 5y + 39 = 0$?
\item[b)\hfill] What distance does $P_1 ~\widehat{=}~ (4, 3)^\top$
                have from the line that intersects the coordinate
                axes at $x = \frac{10}{3}$ and $y = \frac{5}{2}$?
\item[c)\hfill] What distance do the parallels $2x-3y = 6$ and
                $4x-6y = 25$ have?
\end{enumerate}

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\subsubsection*{Exercise 4 \quad\rm Plane Equations}

Which plane through the point $\vec{p} = (-3, 0, 2)^\top$
is perpendicular to the line\\
$\vec{x} = (-1, -2, 0)^\top + k \cdot (1, 1, -1)^\top$
for $k \in \IR$?

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\subsubsection*{Exercise 5 \quad\rm Points on a Plane}

A plane through the point with location vector~$\vec{p}$
that is perpendicular to the vector~$\vec{n}$ has the equation
$(\vec{x} - \vec{p})^\top \vec{n} = 0$.
Let $\vec{p} = (1, -1, 2)^\top$ and $\vec{n} = (1, 2, -3)^\top$. \\
Which of the following points lie in the plane? \\
$P_1 ~\widehat{=}~ (-2, -1, 1)^\top$,
$P_2 ~\widehat{=}~ (1, -1, 2)^\top$,
$P_3 ~\widehat{=}~ (2, -2, 1)^\top$.

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\subsubsection*{Exercise 6 \quad\rm Distance of a Point from a Plane}

Find a formula to compute the distance of a given
point~$\vec{y} = (y_1,y_2,y_3)^{\top}$ to a plane
$(\vec{x}-\vec{p})^{\top} \vec{n} = 0$. Use this formula to compute
the distance for $\vec{p} = (-1,-1,-1)^{\top}$,
$\vec{n} = (4,-2,3)^{\top}$ and $\vec{y} = (3,14,-6)^{\top}$.

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