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{\LARGE Math 240 Algebra I --- Term Test}

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University of Toronto, October 25, 2012

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\noindent{\bf Solve 4 out of the 5 problems on the other side of this page. }\\
Each of the problems is worth 25 points.\\You have an hour and 50
minutes.

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\noindent{\bf Notes.}

\begin{itemize}
\item No outside material other than stationary is allowed.
\item {\bf Neatness counts! Language counts!} The {\em ideal} written
solution to a problem looks like a page from a textbook; neat and clean
and made of complete and grammatical sentences. Definitely phrases like
``there exists'' or ``for every'' cannot be skipped. Lectures are mostly
made of spoken words, and so the blackboard part of proofs given
during lectures often omits or shortens key phrases. The ideal written
solution to a problem does not do that.
\end{itemize}

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\centerline{\bf Good Luck!}

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\noindent{\bf Solve 4 of the following 5 problems. } Each of the problems is
worth 25 points.  You have an hour and 50 minutes. {\bf Neatness counts!
Language counts!}

\vfill \noindent{\bf Problem 1. } Let $F$ be a field.
\begin{enumerate}
\item Prove that for any $a\in F$, we have $0\cdot a=0$.
\item Prove that if $a,b\in F$ and $ab=0$, then either $a=0$ or $b=0$.
\end{enumerate}

\par\noindent{\small {\bf Tip. } Don't start working! Read the whole exam
first. You may wish to start with the questions that are easiest for you.}
 
\vfill \noindent{\bf Problem 2. }
\begin{enumerate}
\item In the field $\bbC$ of complex numbers, compute
\[
  4i(1+i)
  \qquad\text{ and }\qquad
  \frac{4i}{1+i}.
\]
(To be precise, ``compute'' means ``write in the form $a+ib$, where
$a,b\in\bbR$'').
\item In the field $\bbC$ of complex numbers, find an element $z$ so that
$z^2=2i$.
\item In the $11$-element field $F_{11}$ of remainders modulo $11$, find
all solutions of the equation $x^2=-2$.
\end{enumerate}

\vfill \noindent{\bf Problem 3. } Prove that if $W$ is a subspace of a
finite-dimensional vector space $V$, then $W$ is also finite-dimensional.

\vfill \noindent{\bf Problem 4. } Find a polynomial $f\in P_3(\bbR)$ that
satisfies $f(-1)=-3$, $f(0)=0$, $f(1)=1$, and $f(2)=6$.

\vfill \noindent{\bf Problem 5. } Let $V$ and $W$ be vector spaces over the
same field $F$, and let $u$ be some element of $V$. Recall that
$\calL(V,W)$ denotes the vector space of all linear transformations
$L\colon V\to W$.
\begin{enumerate}
\item Define a map $E\colon\calL(V,W)\to W$ by setting $E(L)=L(u)$, for
$L\in\calL(V,W)$. Prove that $E$ is a linear transformation.
\item If in addition $V$ is of dimension 1 and $u\neq 0$, prove that $E$ is
an isomorphism.
\end{enumerate}

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\centerline{\bf Good Luck!}

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