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\begin{document}
\noindent{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1617}{2016-17}:
  \href{http://drorbn.net/?title=1617-257}{MAT 257 Analysis II}:
  \hfill\url{http://drorbn.net/?title=1617-257}
}

\begin{center} {\bf TT2: Q3 and Q6 Marking Key} \end{center}


\noindent{\bf Problem 3. }
\begin{enumerate}
\item Very carefully, define ``a bounded function $f$ is integrable on a rectangle $Q$''. Assume that your reader does not know the words / phrases ``partition of an interval'', ``partition of $Q$'', ``a rectangle in a partition'', ``the volume of a rectangle'', ``lower sum'', ``upper sum'', and whatever remains.
\item Directly from the definitions, prove that constant functions are integrable.
\end{enumerate}

\noindent{\bf Marking Key.} 18 for part 1, of which 9 global and 9 local. 7 for part 2.

$(5/25)$ Fully correct quote/use of ``$f$ integrable iff $\mu(D(f))=0$''.

$(-3)$ The statement $\sum V(R_i)=V(Q)$ is skipped.

$(-4)$ Reverse-order definitions.

$(-2)$ Min/max instead of inf/sup.

$(-1)$ Restricted to 2D.

\vfill\noindent{\bf Problem 6. } Let $x_1,\ldots,x_k\in\bbR^n$. Prove:
\begin{enumerate}
\item Interchanging two of these vectors does not change the volume of the parallelepiped they span.
\item Adding a multiple of one of those vectors to another one does not change the volume of the parallelepiped they span.
\item Multiplying one of these vectors by a scalar $\lambda\in\bbR$ multiplies the volume of the parallelepiped they span by $|\lambda|$.
\item If these vectors are orthonormal, the volume of the parallelepiped they span is $1$.
\end{enumerate}

\noindent{\small {\bf Tip. } For this problem, you may freely use everything stated in class.}

\vskip 3mm
\noindent{\bf Marking Key.} 5 points for using a reasonable volume formula or definition. Then 5 points for each part.

$6/25$ if pretended that $k=n$.

$(-15)$ All hinges on $\det(AB)=\det(A)\det(B)$ for non-square matrices.

$(-2)$ The action on $X^TX$ is just on rows/columns, rather than both.

$(-3)$ Column actions by $X\mapsto EX$, then cancelled middle terms in $\det(X^TE^TEX)$.

\vfill

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