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\title{MA20008 Algebra 1, 2004, Sheet 5}
\author{Geoff Smith, {\tt http://www.bath.ac.uk/$\sim$masgcs}}
\date{}
\begin{document}
\maketitle
\begin{enumerate}
\item Let $V$ be a vector space of dimension
$n$. 
\begin{enumerate}
\item Suppose that we have subspaces
\[ 0 = V_0  < V_1 < V_2 \cdots < V_n = V\]
with $\dim V_i = i$ for every $i = 0,1,\ldots, n$.
For each $i>0$ choose ${\bf v_i} \in V_i$ but ${\bf v_i} \not \in V_{i-1}$.
Show that 
\[ {\bf v_1}, {\bf v_2}, \ldots, {\bf v_n}\]
is a basis of $V$.
\item Suppose that
\[ {\bf v_1}, {\bf v_2}, \ldots, {\bf v_n}\]
is a basis of $V$. Show that there are subspaces
\[ 0 = V_0  < V_1 < V_2 \cdots < V_n = V\]
with $\dim V_i = i$ for every $i = 0,1,\ldots, n$ such that
${\bf v_i} \in V_i$ but ${\bf v_i} \not \in V_{i-1}$
for every $i >0$. 
\end{enumerate}

\item Let $V$ be a vector space of dimension 
$n$. Suppose that $\alpha: V \longrightarrow V$ is a linear
map.
\begin{enumerate}
\item Show that $V \geq \mbox{ Im\ }\alpha \geq \mbox{ Im\ }\alpha^2
\geq \ldots \geq 0$.   {\em Observation: $\mbox{ Im\ }\alpha^r$
denotes the map defined by composing $r$ copies of $\alpha$.}
\item Show that $0 \leq \mbox{ Ker }\alpha \leq \mbox{ Ker }\alpha^2
\leq \cdots \leq V$. 
\item Suppose that $r$ is a natural number and 
$\mbox{ Im\ }\alpha^r = \mbox{ Im\ }\alpha^{r+1}$. Show
that $\mbox{ Im\ }\alpha^t = \mbox{ Im\ }\alpha^r$ for all
natural numbers $t \geq r$.
\item Let $V$ be a vector space of dimension
$n$.  Suppose that $\alpha: V \longrightarrow V$ is a linear
map such that $\alpha^2 = 0$ (the zero map). Show that
the nullity $\nu_\alpha$ satisfies $2\nu_\alpha \geq n$.
\item Suppose that there is a natural number $m$ such that
$\alpha^m$ is the zero map $0$. Prove that 
$\alpha^n$ is $0$.
\end{enumerate}
\item Suppose that $V$ is a vector space of dimension
$n$ and that $\alpha: V \longrightarrow V$ is a linear
map. Suppose that $\alpha^3$ is $0$, the zero map.
Show that $\nu_\alpha \geq n/3$ where $\nu_\alpha$
denotes the nullity of $\alpha$.
{\rm Hint: let $V_0 = V,\ V_1 = \mbox{ Im }\alpha$ 
and $V_2 = \mbox{ Im }\alpha^2$.
Let $\alpha_1 : V \longrightarrow V_1$ and 
$\alpha_2 : V_1 \longrightarrow V_2$ be the maps defined by 
$\alpha_i( {\bf v}) = \alpha( {\bf v})$ for all ${\bf v} \in V_{i-1}.$
Deploy the rank nullity theorem.}
\item Let $V$ be a finite dimensional vector space, with
$W, X, Y$ and $Z$ all subspaces of $V$.
\begin{enumerate}
\item Show that 
\begin{eqnarray*}
& &  \dim (X \cap Y) + \dim ((X+Y) \cap Z)\\ 
& = & \dim (Y \cap Z) + \dim ((Y+Z) \cap X)\\
& = & \dim (Z \cap X) + \dim ((Z+X) \cap Y).
\end{eqnarray*} 
\item Show that
\begin{eqnarray*}
& &  \dim (W \cap X) + \dim (Y \cap Z) + \dim((W + X) \cap (Y + Z))\\
&=&  \dim (W \cap Y) + \dim (X \cap Z) + \dim((W + Y) \cap (X + Z))\\
&=&  \dim (W \cap Z) + \dim (X \cap Y) + \dim((W + Z) \cap (X + Y))
\end{eqnarray*}
\end{enumerate}
{\em Hint: this question is not hard. You need an idea. Please do
not start picking bases; that way madness lies.}
\end{enumerate}
\end{document}
