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\begin{document}

\title{Mathematics for Data Science}
\author{Final Exam}
\date{February 19, 2026}
\maketitle
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\begin{enumerate}

\item Consider the subspace $V \subset C([0,2\pi];\C)$ spanned by
$B=[1,\e^{\i x},\e^{-\i x}]$ and endowed with inner product
\[
  \langle f, g \rangle = \frac1{2\pi} \int_0^{2 \pi} \overline{f(x)}
    \, g(x) \, \d x \,.
\]
When answering the following, Euler's formula
\[
  \e^{\i z} = \cos z + \i \, \sin z
\]
may be helpful.
\begin{enumerate}
\item Verify that $\langle \,\cdot\,,\,\cdot\,\rangle$ is an inner
product.
\item Show that the basis $B$ is orthonormal.  To limit the number of
required computations, full credit if you check the inner products
$\langle 1, 1 \rangle$, $\langle 1, \e^{\i x} \rangle$, $\langle
\e^{\i x}, \e^{\i x} \rangle$, and $\langle \e^{-\i x}, \e^{\i x}
\rangle$.
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\item Consider the linear operator $Lp = p''$.  State nullspace and
range of $L$; no computation required.  Then 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 Show that $B'=[1,\cos x, \sin x]$ is also a basis for $V$ by
writing down the matrix representing the change of basis from $B$ to
$B'$.
\item Find the matrix representing $L$ with respect to basis $B'$.
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\item Let $\ell \colon V \to \C$ be defined by $\ell(f) = f'(0)$.
Show that this is a linear transformation.
\item Find a function $h \in V$ such that for every $f \in V$,
\[
  \ell(f) = \langle h, f \rangle \,.
\]
\end{enumerate}
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\item Are the following statements true or false?  Give a \emph{brief}
explanation in each case.  No credit for a true/false guess without
explanation!
\begin{enumerate}
\item Every metric space has a basis.
\item The closure of a set can be open.
\item The unit ball in $\R^n$ is compact.
\item The unit ball in $L^2([0,1];\R)$ is compact.
\item In a Banach space, every Cauchy sequence converges.
\end{enumerate}
\hfill(2+2+2+2+2)


\item Show that the function
\[
  f(x,y) =
  \begin{cases}
    \dfrac{xy}{x^2+y^2} & \text{for } (x,y) \neq (0,0) \\
    0 & \text{for } (x,y) = (0,0)
  \end{cases}
\]
is not differentiable at the origin.
\hfill(5)


\item
\begin{enumerate}
\item Find the second-order Taylor polynomial of the function
\[
  f(x,y) = \e^{x+y^2}
\]
at the point $(0,0)$.

\item On $V=M_n(\F)$, consider the (nonlinear) map
\[
  f(A) = A^{-1}
\]
Show that when $A$ is invertible,
\[
  Df(A)B = -A^{-1}BA^{-1} \,.
\]
\end{enumerate}
\hfill(5+5)


\item Let $X$ be a Banach space and $B \in \mathcal{B}(X)$ a bounded
linear operator on $X$ with operator norm $\lVert B \rVert < \frac12$.

Consider the map $f \colon \mathcal{B}(X) \to \mathcal{B}(X)$ defined
by
\[
  f(A) = ABA
\]
\begin{enumerate}
\item Show that $f$ is not a linear transformation but $f(A)$ is a
linear transformation.  (On which space?)
\item Show that $f$ maps the closed unit ball in $\mathcal{B}(X)$ into
itself.
\item Show that $f$ has a fixed point on the closed unit ball in
$\mathcal{B}(X)$.
\end{enumerate}
\hfill (5+5+5)


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



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