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\begin{document}

\begin{center}
{\bf \Large 250A Homework 1}\\[7mm]
{\small Due Monday October 11}\\[4mm]
\end{center}

\noindent
{\bf Question 1} 
Let $T:\Real^3\rightarrow\Real^3$ be a linear operator.
Show that $T$ has 1 and 2 dimensional invariant
subspaces.

\vspace{.4cm}\noindent
{\bf Question 2} 
Let $M$ be the space of $3\time 3$ matrices. What is $\dim(M)$?
Now define the linear operator $T:M\rightarrow M$ by
\be
M\ni m\stackrel{T}{\longmapsto}
\frac12\left[\begin{pmatrix}1&&\\&2&\\&&1\end{pmatrix}\ m
+ m \ \begin{pmatrix}1&&\\&2&\\&&1\end{pmatrix}\right]\, .
\ee
Compute $\det T$.

\vspace{.4cm}\noindent
{\bf Question 3} {\it Van der Monde Determinant.} Let $A$ be the $(n\times n)$
matrix with entries
\be
A_{ij}=(x_i)^{j-1}\, .
\ee
Show that $\det A=\prod_{i>j}(x_i-x_j)$.
 
\vspace{.4cm}\noindent
{\bf Question 4} 
{\it (Anti)commutators.}
Let $V$ be a finite dimensional vector space. Show that
the mapping
\be
[\cdot,\cdot]:L(V)\times L(V)\rightarrow L(V)
\ee
where
\be
[\cdot,\cdot]:(M,N)\mapsto MN-NM \equiv [M,N]\, ,
\ee
obeys the Leibnitz rule $[M,NR]=[M,N]R+N[M,R]$.
In addition, verify the Jacobi identity
\be
[M,[N,R]]+[N,[R,M]]+[R,[M,N]]=0\, .
\ee
Generalize the above laws to the mapping $\{\cdot,\cdot\}:
(M,N)\mapsto MN+NM\equiv\{M,N\}$. Include also new rules
which mix both operations.


\vspace{.4cm}\noindent
{\bf Question 5} 
{\it Baker Campbell Hausdorff Formula.}
Let $V$ be a finite dimensional vector space and $M,N\in L(V)$.
Show that 
\be
\exp(M+N)=\exp(M)\exp(N)\exp(-\frac12[M,N])\, ,
\ee
if $0=[M,[M,N]]=[N,[M,N]]$.
{\it Hint: Develop and solve a differential equation for
$R(\lambda)\equiv\exp(\lambda M)\exp(\lambda N)\in L(V)$}.
 
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