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

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University of Toronto, October 24, 2006

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\noindent{\bf Solve the 5 problems on the other side of this page. }\\
Each of the problems is worth 20 points.\\You have an hour and 45
minutes.

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\noindent{\bf Notes.}

\begin{itemize}
\item No outside material other than stationary and a basic calculator
is allowed.
\item We will have an extra hour of class time in our regular class
room on Thursday, replacing the first tutorial hour.
\item The final exam date was posted by the faculty --- it will take place
on Wednesday October 13 from 2PM until 5PM at room 3 of the Clara Benson
Building, 320 Huron Street (south west of Harbord cross Huron, home of the
Faculty of Physical Education and Health).
\end{itemize}

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\centerline{\bf Good Luck!}

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\noindent{\bf Solve the following 5 problems. } Each of the 
problems is worth 20 points.  You have an hour and 45 minutes.

\vfill \noindent{\bf Problem 1. } Let $F$ be a field with zero element
$0_F$, let $V$ be a vector space with zero element $0_V$ and let $v\in V$
be some vector. Using only the axioms of fields and vector spaces, prove
that $0_F\cdot v=0_V$.

\vfill \noindent{\bf Problem 2. } 
\begin{enumerate}
\item In the field $\bbC$ of complex numbers, compute
\[
  \frac{1}{2+3i}+\frac{1}{2-3i}
  \qquad\text{ and }\qquad
  \frac{1}{2+3i}-\frac{1}{2-3i}.
\]
\item Working in the field $\bbZ/7$ of integers modulo 7, make a table
showing the values of $a^{-1}$ for every $a\neq 0$.
\end{enumerate}

\vfill \noindent{\bf Problem 3. } Let $V$ be a vector space and let $W_1$
and $W_2$ be subspaces of $V$. Prove that $W_1\cup W_2$ is a subspace of
$V$ iff $W_1\subset W_2$ or $W_2\subset W_1$.

\vfill \noindent{\bf Problem 4. } In the vector space $M_{2\times
2}(\bbQ)$, decide if the matrix $\begin{pmatrix}1&2\\-3&4\end{pmatrix}$ is
a linear combination of the elements of
$S=\left\{\begin{pmatrix}1&0\\-1&0\end{pmatrix},\
\begin{pmatrix}0&1\\0&1\end{pmatrix},\
\begin{pmatrix}1&1\\0&0\end{pmatrix}\right\}$.

\vfill \noindent{\bf Problem 5. } Let $V$ be a finite dimensional
vector space and let $W_1$ and $W_2$ be subspaces of $V$ for which
$W_1\cap W_2=\{0\}$. Denote the linear span of $W_1\cup W_2$ by $W_1+W_2$.
Prove that $\dim(W_1+W_2)=\dim W_1 + \dim W_2$.

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\centerline{\bf Good Luck!}

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