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\lhead{\bf Math 251 Final Exam}
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\rhead{\bf August, 2012}
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{\Large \bf McGill University}\hfill {\Large August 2012}\medskip
{\Large \bf Faculty of Science}\hfill {\Large Supplemental Final examination}
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\begin{center}{\Large \bf Abstract Algebra II}%insert title of course
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{\large \bf Math 251}%Insert course number
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???day, August ??, 2012 %Insert Date
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Time: ?:00 am to ??:00 pm %Insert Time
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\noindent {\large Examiner: Prof. H. Darmon \hfill Associate Examiner: Prof. E.Z.Goren}%Insert names
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{\bf INSTRUCTIONS}%Put in the instructions
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\begin{enumerate}[1.]
\item Please write your answers \underline{clearly} in the exam booklets provided.
\item You may quote any result/theorem seen in the lectures or in the assignments without
proving it (unless, of course, it is what the question asks you to prove).
\item This is a closed book exam.
\item Translation dictionary is permitted.
\item Calculators are not permitted.
\end{enumerate}%Check the instructions
\bigskip\noindent This exam comprises the cover page and two
pages of questions, numbered 1 to 10. Each question is worth 10 points.
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%{\Huge 189-251A: Algebra 2} \\ \sk
%{\Huge Supplemental Final Exam} \\ \sk
%{\Large August, 2012 }
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\sk\sk\sk
\noindent
1. a) Let $F$ be a field, and let $V$ be the set of $m\times n$ matrices
with entries in $F$.
Explain (briefly) why $V$ is a vector
space over $F$, and
compute its dimension by exhibiting an
{\em explicit basis } for $V$.
\sk
b) Suppose that $V$ is the set of all (infinite) sequences with values in $F$,
having only finitely many non-zero values. (I.e., a typical element of $V$ is a sequence $(a_1,a_2,\ldots, a_n,\ldots)$
with $a_i\in F$ and such that $a_i=0$ for all $i\ge M$ for a suitable $M$ depending in general on the sequence $(a_i)$.) Does
$V$ have a countable basis? Justify
your answer.
\sk\sk
\sk\noindent
2. A {\em generalised Fibonnaci sequence} is a sequence $(a_0, a_1, \ldots, a_n,\ldots)$ of
real numbers satisfying the recursive relation
$$ a_{n+1} = a_{n+1} + a_n, \qquad \mbox{ for all } n\ge 0.$$
Show that the set of such sequences is a vector space over the field
${\bf R}$ of real numbers, and compute its dimension.
\sk
\sk
\sk\noindent
3. a) Let $f(x)$ be a polynomial with coefficients in a field $F$, and let
$V = F[x]/(f(x))$ be the quotient of the polynomial ring $F[x]$
by the ideal generated by $f(x)$, viewed as a vector space over $F$.
Let
$T: V\lra V$ be the linear transformation given by
$T(p(x)) = (x+1) p(x)$.
\sk
a) Compute the minimal polynomial of $T$.
\sk
b) Compute the characteristic polynomial of $T$.
\sk
c) Give a necessary and sufficient condition on $f(x)$ for $T$ to be diagonalisable over $F$.
\sk
d) Give a necessary and sufficient conditon on $f(x)$ for $T$ to be
invertible.
\sk\sk\sk\noindent
4. A {\em projection} is a linear transformation (on a finite-dimensional
vector space) satisfying the relation $T^2=T$.
Show that a projection is always diagonalisable.
\sk\sk
\sk\noindent
5. Let $V$ be the real vector space of all functions of the form
$g(x) e^x$, where $g$ is a real polynomial of degree $\le 3$,
and let $T:V\lra V$ be the linear transformation given by
$$ T(f) = f+2f'.$$
Write down a basis for $V$ and the matrix of $T$ relative to this basis.
\newpage
\sk\sk
\sk\noindent
6. Define the following terms:
\sk
a) The {\em dual space} $V^*$ of a vector space $V$;
\sk
b) The {\em dual linear map} $T^*$ attached to a
linear transformation $T:V\lra W$. Be sure to specify what the domain and
target of $T^*$ are, and to write down the formula defining $T^*$.
\sk
c) Show that $(T_1 T_2)^* = T_2^* T_1^*$ for all
$T_1: V\lra W$ and $T_2:U\lra V$.
\sk\sk\sk\noindent
7. Let $V$ be the vector space $F^7$ of $7$-tuples of elements of a field $F$,
and let $W$ be the kernel of the map $M: F^7\lra F^3$
described by the following $3\times 7$ matrix (relative to the
standard bases of course)
$$ M = \left( \begin{array}{ccccccc}
0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 1 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 & 0 & 1 \end{array}\right),
$$
Show that two vectors in $V$ having all but {\em exactly one} coordinate in common cannot both belong to $W$.
\sk\sk\sk\noindent
8. Let ${\bf R}^3$ be equipped with the standard inner product.
Starting with the basis $(1,1,0)$, $(2,2,3)$, $(3,-1,2)$ of
${\bf R}^3$, compute the orthonormal basis that is derived from it by performing
the Gramm-Schmidt orthogonalisation process.
\sk\sk
\sk\noindent
9. A linear transformation $T$ on a
finite-dimensional Hermitan inner product space
is said to be {\em self-adjoint} if it satisfies the relation
$T^*=T$.
\sk
a) If $v_1$ and $v_2$ are eigenvectors for $T$ associated to {\em distinct} eigenvalues, show that they are orthogonal.
\sk
b) Show that a self-adjoint $T$ has an eigenvalue, and that all its eigenvalues are real.
\sk\sk
\sk\noindent
10.
Let $V=\R^3$ equipped with the standard dot product and resulting
distance function, and let $W$ be the hyperplane (i.e.,
subspace of dimension $2$) defined by the equation
$$ W=\{ (x,y,z) \mbox{ with } x + y = 2z \}.$$
Compute the vector in $W$ which is closest to the vector
$(1,1,1)$, relative to the standard Euclidean distance.
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