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\begin{center}
  {\Large UNIVERSITY OF TORONTO}\\
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  {\large Faculty of Arts and Sciences}\\
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  {\Large DECEMBER EXAMINATIONS 2012}\\
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  {\Large Math 240H1 Algebra I --- Final Exam}

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  Dror Bar-Natan\par
  December 13, 2012\par
\end{center}

\vfil

Solve all of the following 5 questions. The questions carry equal weight
though different parts of the same question may be weighted differently.

\vfil

\noindent{\bf Duration. } You have 3 hours to write this exam.

\vskip 2mm

\noindent{\bf Allowed Material. } Basic calculators, not capable of
displaying text or sounding speech.

\vfil

\centerline{{\bf Good Luck!}}

\vfil

\newpage 

\noindent{\bf Problem 1. } It is given that $W_1$ and $W_2$ are subspaces
of the same vector spaces $V$. Prove that their union $W=W_1\cup W_2$ is
also a subspace of $V$ if and only if $W_1\subset W_2$ or $W_2\subset W_1$.

\vskip 3mm

\par\noindent{\small {\bf Tip. } ``If and only if'' means that 
there are two things to prove.}

\vskip 3mm

\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.}

\vfil\noindent{\bf Problem 2. } Prove the ``replacement lemma'': Let $G$
be a set of $g$ vectors that spans some vector space $V$ and let $L$
be some set of $l$ linearly independent vectors in $V$ (where $g$ and
$l$ are both finite).  Then $g\geq l$ and there is a subset $R$ of $G$,
consisting of $r:=g-l$ vectors, so that $\operatorname{span}(R\cup L)=V$.

\vskip 3mm

\par\noindent{\small {\bf Tip. } As always in math exams, when
proving a theorem you may freely assume anything that preceded it but
you may not assume anything that followed it.}

\vskip 3mm

\par\noindent{\small {\bf Tip. } Neatness, cleanliness and
organization count, here and everywhere else!}

\vfil\noindent{\bf Problem 3. } Recall that the real numbers $\bbR$ are a
vector space over the field $\bbQ$ of rational numbers. Let $V$ be the
subspace of $\bbR$ given by
$V=\{a+b\sqrt{3}\colon a,b\in{\mathbb Q}\}$. 
\begin{enumerate}
\item Find a basis $\beta$ for $V$ over $\bbQ$.
\item Let $T\colon V\to V$ be the linear operator defined by
$Tx=\sqrt{3}x$. Find the matrix representing $T$ relative to the basis
$\beta$ you found in the previous part of this question.
\end{enumerate}

\par\noindent{\small {\bf Tip} (added after exam). Note that $\beta$ is the
basis of both the domain and the target space of $T$.}

\newpage

% Table[i*j, {i, 0, 4}, {j, 0, 4}] // MatrixForm // TeXForm
% Table[i+j, {i, 0, 4}, {j, 0, 4}] // MatrixForm // TeXForm
\noindent{\bf Problem 4. } Let $M$ be the $5\times 5$
``multiplication table'' matrix shown below, let $A$ be the $5\times 5$ 
``addition table'' matrix shown below, and let $S$ be the $6\times 6$
``snakes and ladders'' matrix shown below:
\[ \begin{array}{ccccc}
\left(
\begin{array}{ccccc}
 0 & 0 & 0 & 0 & 0 \\
 0 & 1 & 2 & 3 & 4 \\
 0 & 2 & 4 & 6 & 8 \\
 0 & 3 & 6 & 9 & 12 \\
 0 & 4 & 8 & 12 & 16 \\
\end{array}
\right)
&&
\left(
\begin{array}{cccccc}
\begin{array}{ccccc}
 0 & 1 & 2 & 3 & 4 \\
 1 & 2 & 3 & 4 & 5 \\
 2 & 3 & 4 & 5 & 6 \\
 3 & 4 & 5 & 6 & 7 \\
 4 & 5 & 6 & 7 & 8 \\
\end{array}
\end{array}
\right)
&& \left(\begin{array}{cccccc}
 36 & 35 & 34 & 33 & 32 & 31 \\
 25 & 26 & 27 & 28 & 29 & 30 \\
 24 & 23 & 22 & 21 & 20 & 19 \\
 13 & 14 & 15 & 16 & 17 & 18 \\
 12 & 11 & 10 & 9 & 8 & 7 \\
 1 & 2 & 3 & 4 & 5 & 6 \\
\end{array}\right) \\
\\
M & & A & & S
\end{array} \]
\begin{enumerate}
\item Bring the matrices $M$ and $A$ to reduced row echelon form.
\item Determine the ranks of $M$ and of $A$.
\item Show that every row of the matrix $S$ is a linear combination of its
bottom row and the row $(1\ 1\ 1\ 1\ 1\ 1)$.
\item Deduce that the rank of $S$ is at most $2$.
\item Show that the rank of $S$ cannot be $0$ or $1$, and hence it must be
$2$.
\end{enumerate}

\par\noindent{\small {\bf Note} (added after exam). People preferred
computation and didn't quite get the point of determining the rank of $S$ by
the 3--5 sequence resulting in a harder-to-grade question.}

\vfil\noindent{\bf Problem 5. } Let $A$ be the matrix $A=\left(
\begin{array}{cc} 0 & 1 \\ -2 & 3 \\ \end{array} \right)$.
\begin{enumerate}
\item Compute $\det(A-\lambda I)$.
\item Find the eigenvalues $\lambda_1$ and $\lambda_2$ of $A$.
\item Find their corresponding eigenvectors $v_1$ and $v_2$.
\item Find a matrix $C$ for which $AC=CD$, where $D=\left(
\begin{array}{cc} \lambda_1 & 0 \\ 0 & \lambda_2 \\ \end{array} \right)$.
\item Compute the inverse of $C$.
\item Compute $A^7$ by computing $CD^7C^{-1}$.
\end{enumerate}

\par\noindent{\small {\bf Note} (added after exam). Instead of $A^7$, I
should have asked for a general formula for $A^n$, for $n\in{\mathbb N}$.}

\vfil \centerline{\bf Good Luck!} \vfil

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