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\begin{document}

\title{Mathematics for Data Science}
\author{Mock Exam}
\date{January 26, 2026}
\maketitle
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\begin{enumerate}

\item Consider the subspace $V \subset C([0,1];\R)$ spanned by the
basis $B=[1,\e^x]$ and endowed with inner product
\[
  \langle p, q \rangle = \int_0^1 p(x) \, q(x) \, \d x \,.
\]
\begin{enumerate}
\item Consider the linear operator $Lp = p'$.  State nullspace and
range of $L$; no computation required.
\item State the rank-nullity theorem and show that it applies in this
example. 
\item Find the matrix representing $L$ with respect to basis $B$.
\item Find an orthonormal basis for $V$.
\item Find the matrix representing the change of coordinates from
basis $B$ to your orthonormal basis.
\item Find the matrix representing $L$ with respect to your
orthonormal basis.
\end{enumerate}
\hfill(2+2+2+3+3+3)

\item Let $V=M_2(\R)$.  Fix two vectors $u, v \in \R^2$ and define
\[
  f(A) = u^T A v \,.
\]
\begin{enumerate}
\item Show that $f \colon V \to \R$ is a linear transformation.
\item Recall the definition of the Frobenius inner product on $V$,
\[
  \langle A, B \rangle =
  \sum_{i,j=1}^n a_{ij} \, b_{ij} = 
  \operatorname{tr} (A^TB) \,.
\]
and set
\[
  u =
  \begin{pmatrix}
  1 \\ 1
  \end{pmatrix} \,,
  \qquad v = 
  \begin{pmatrix}
  2 \\ -1
  \end{pmatrix} \,.
\]
Find a matrix $B \in V$ such that
\[
  f(A) = \langle A,B \rangle \,.
\]  
\end{enumerate}
\hfill(5+5)



\item Let $V=M_n(\F)$ with $n\geq 2$.  The determinant is a map $\det \colon V \to \F$.
\begin{enumerate}
\item Show that $\det$ is not a linear transformation.
\item Give an argument that $\det$ is differentiable.

\emph{Hint:} Computing the derivative is possible, but it is not easy.
Better use a short (!) general argument.

\item Show that $D \det(0) = 0$.

\emph{Note:} This formula is using the symbol $0$ in two different
meanings.  As part of your answer, you should state explicitly what
each of the zero symbols means, i.e., which space it belongs to.
\end{enumerate}
\hfill(5+5+5)

\item Are the following statements true or false?  If true, state a
brief explanation or name the relevant theorem.  A full proof is not
required.  If false or incomplete, modify the statement so that it
becomes true, and give a short comment on your modification.
\begin{enumerate}
\item A normed vector space is a metric space.
\item A set can be both open and closed.
\item A set is compact if it is closed and bounded.
\item A sequence, together with its limit, is a closed set.
\item The rational numbers are a closed set.
\end{enumerate}
\hfill(2+2+2+2+2+2)

\item
\begin{enumerate}
\item Find the second-order Taylor polynomial of the function
\[
  f(x,y) = (2-x-y)^3
\]
at the point $(1,1)$.

\item On $C([0,\infty),\R)$, consider the (nonlinear) map
\[
  T(f)(t) = \int_0^t \ln f(s) \, \d s \,.
\]
Compute the derivative $DT(f)[g](t)$.

\emph{Note:} A formal computation suffices -- you do not need to prove
that your result indeed satisfies the properties of the Fr\'echet
derivative.  You may assume that $f(t) \geq 1$ for all
$t \in [0,\infty)$.
\end{enumerate}
\hfill(5+5)

\item Let $X$ be a Banach space, $A \colon X \to X$ a linear operator
with operator norm $\lVert A \rVert < 1$, and $b \in X$.

Show that
$f(x) = Ax+b$ has a fixed point.
\hfill (10)

\item Let $f \colon \R^3 \to \R^2$ be a $C^1$-function such that
$f(5,-2,1)=0$ and
\[
  Df(5,-2,1)
  =
  \begin{pmatrix}
    1 & -1 & 2 \\
    1 & 1 & -2
  \end{pmatrix} \,.
\]
\begin{enumerate}
\item Show that there exists an open interval
$I = (1-\delta,1+\delta)$ and a differentiable function
$g \colon I \to \R^2$ such that $g(1) = (5, -2)$ and
$f(g_1(z), g_2(z), z) = 0$ for all $z \in I$.
\item Compute $Dg(1)$.
\item Does there exist a differentiable function $h$ defined on some
neighborhood of $x=5$ with values in $\R^2$ such that
$f(x,h_1(x), h_2(x))=0$?
\end{enumerate}
\hfill(5+5+5)




\end{enumerate}



\end{document}