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{\Huge 189-251A: Algebra 2} \\ \sk
{\Huge Final Exam} \\ \sk
{\Large Monday, April 23, 2012 }
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{\em
This exam has 10 questions, worth 10 points each.
Calculators and class notes are not allowed. }
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1. a) Let $F$ be a field, let $S$ be a finite set and let $V$ be the vector
space of
functions from $S$ to $F$. Compute the dimension of $V$ by exhibiting an
{\em explicit basis } for $V$.
b) Suppose $S$ is infinite, but countable. Does
$V$ then have a countable basis? Justify
your answer.
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2. Let $p(x)$ be a non-zero
polynomial of
degree $n$ with coefficients in a field $F$
and let $V$ be the quotient $F[x]/(p(x))$.
Compute the dimension of $V$ by exhibiting an explicit basis for
$V$. (You should include a proof that the set of vectors
you come up with is indeed a basis...)
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3. a) Let $V$ be the vector space
of problem $2$, and let
$T: V\lra V$ be the linear transformation given by
$T(p(x)) = x p(x)$.
a) Compute the minimal polynomial of $T$.
b) Compute the characteristic polynomial of $T$.
c) Give a necessary and sufficient condition on $p(x)$ for $T$ to be diagonalisable over $F$.
d) Give a necessary and sufficient conditon on $p(x)$ for $T$ to be
invertible.
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4. Give an example of a non-zero vector space $V$ and
a linear transformation $T:V\lra V$ satisfying $\ker(T)={\rm image}(T)$.
Show that such a linear transformation is {\em never} diagonalisable.
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5. Let $V$ be the vector space of $2\times 2$ matrices with entries in
the field $\R$ of real numbers,
and let $T:V\lra V$ be the linear transformation given by
$$ T(M) = A M A^{-1}, \qquad A =\left(\begin{array}{cc}
{1} & {2} \\ 0 & {1}\end{array} \right).$$
Write down a basis for $V$ and the matrix of $T$ relative to this basis.
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6. Let $T:V\lra V$ be a diagonalisable linear transformation, let
$\lambda_1,\ldots,\lambda_t$ be the distinct eigenvalues for $T$ and let
$$ V= \oplus_{i=1}^t V_{\lambda_i}$$
be the associated decomposition of $V$ into a direct sum of eigenspaces.
Show that a linear transformation $U:V\lra V$ commutes with $T$
{\em if and only if} all the eigenspaces $V_{\lambda_i}$ are stable under $U$.
(I.e., if and only if $U$ maps $V_{\lambda_i}$ to itself, for each $i$.)
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7. Define the following terms:
a) The {\em dual space} $V^*$ of a vector space $V$;
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 $T^*$.
c) Show that $(T_1 T_2)^* = T_2^* T_1^*$ for all
$T_1: V\lra W$ and $T_2:U\lra V$.
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8. State and prove the Cauchy-Scwartz inequality for real inner product spaces.
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9. A linear transformation $T$ on a
finite-dimensional Hermitan inner product space
is said to be {skew-adjoint} if it satisfies the relation
$T^*=-T$.
a) Show that a skew-adjoint operator is normal.
b) Show that all the eigenvalues of a skew-adjoint operator are purely
imaginary.
c) Show that every normal operator $T$ can be written as a sum
$T_1 + T_2$ where $T_1$ is self-adjoint, $T_2$ is skew-adjoint,
and $T_1T_2 = T_2 T_1$.
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10. Let $V=\R^n$ equipped with the standard dot product and resulting
distance function, and let $W$ be the hyperplane (i.e.,
subspace of dimension $n-1$) defined by the equation
$$ W=\{ (x_1,\ldots,x_n) \mbox{ with } x_1+\cdots + x_n =0 \}.$$
Show that the vector in $W$ which is closest to the vector
$(x_1,\ldots, x_n)$ is the vector
$(x_1-\mu,\ldots,x_n-\mu)$, where
$\mu := \frac{x_1+\cdots + x_n}{n}$ is the {\em mean}
of $x_1,\ldots,x_n$.
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