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{\Huge 189-251B: Algebra 2} \\ \vskip 0.1in
{\Huge Assignment 5} \\ \vskip 0.1in
{\Large Due: Wednesday, February 12. }
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1. Suppose that $T:V\lra V$ is a linear transformation of vector spaces
over $\R$ whose minimal polynomial has no multiple roots.
Show that $V$ can be expressed as a direct sum
$$ V = V_1\oplus V_2 \oplus\cdots \oplus V_t$$
of $T$-stable subspaces of dimensions at most $2$.
Show that, relative to a suitable basis, $T$ can be represented by an
$n\times n$ matrix with at most $2n$ non-zero entries, where $n:=\dim(V)$.
\sk\noindent
2. Let $g(x)$ be a polynomial in $F[x]$ of degree $d$ and let
$V$ be the ring $F[x]/(g(x))$.
(a) Show that $V$ is a $d$-dimensional
vector space over $F$.
(b) Let $T$ be the function on $V$ defined by
the rule $ T(v) = [x] v$. (Where $[x]$ denotes the equivalence class
of $x$ in the quotient ring.)
What is the minimal polynomial of $T$?
When is $T$ diagonalisable?
\sk\noindent
3. Let $A$ be an upper-triangular matrix with entries in a field $F$.
Show that $A$ is diagonalisable if all its diagonal entries are distinct.
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4. Let $A$ be an upper-triangular matrix with entries in a field $F$.
Suppose that all the diagonal entries of $A$ are equal.
Show that $A$ is diagonalisable if and only if it is diagonal.
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5. Let $A$ be an invertible matrix with entries in $F=\Z_p$. Show that $A$ is
diagonalisable if and only if its order (i.e., the least $t$ with
$A^t = 1$ in the group ${\bf GL}_n(\Z_p)$ of $n\times n$ matrices with
entries in $\Z_p$) divides $p-1$.
\sk\noindent
6. Let $V$ be the (infinite-dimensional) vector space of
infinitely differentiable real-valued functions on the real line,
and let $W$ be the subset of $V$ consisting of functions $f$
satisfying the
differential equation
$$ a_n f^{(n)} + a_{n-1} f^{(n-1)} + \cdots + a_1 f' + a_0 f = 0,$$
where the scalars $a_j$ are real. (Here, $f^{(j)}$ denotes the
$j$-th derivative of $f$.)
(a)
Show that $W$ is a vector subspace of $V$. (This is why this equation is
called a {\em linear} differential equation!)
(b) Let $T$ be the function from $V$ to $V$ defined by $T(f) = f'$.
Show that $T$ is a linear transformation that maps $W$ to itself.
(c) From now on, use $T$ to denote the restriction of $T$ to the space
$W$. Show that the eigenvalues of $T$ are precisely the roots of
the polynomial $ g(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$.
(d) Show that the eigenspace for $T$ attached to an eigenvalue $\lambda$ is
{\em always} one-dimensional.
(e) Suppose that $(x-\lambda)^r$ divides $g(x)$ exactly (i.e., no higher power
of $(x-\lambda)$ divides it).
Show that the generalised eigenspace for $T$ attached to the eigenvalue $\lambda$ is
of dimension $r$ and write down a basis $f_1,\ldots, f_r$
for this generalised eigenspace such that $f_j$ belongs to the kernel of $(T-\lambda)^j$
but not of $(T-\lambda)^{j-1}$.
(You may use without proof all that you know about
linear differential equations of first order.)
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