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{\left[\begin{smallmatrix}}
{\end{smallmatrix}\right]}

\title{Math 23a 1st Midterm exam, Oct~19}
\dedicatory{Good luck!}

\begin{document}
\maketitle

There are 50 points total; further breakdown is by question.

\begin{enumerate}
\item (8 points) 
Let $A=\begin{bsmallmatrix}1&1&0\\0&1&1\\0&0&1\end{bsmallmatrix}$. Find all
$3\times 3$ matrices such that $AB=BA$.

\item (10 points) Let $V=\bR^3, \set{e_1,e_2,e_3}$ be the standard basis of $V$,
and let $v_1= e_1+e_2,v_2=e_2 +e_3,v_3=e_3$.
Let $\set{e^1 ,e^2,e^3},\set{ v^1 ,v^2,v^3}\subset V^\vee=V^*$ be the
dual bases.
Write $v^2$ as a linear combination of $e^1 ,e^2,e^3$.
 
\item (12 points) Let $\bC$ be the set of complex numbers. 
\begin{enumerate}
 \item (3 points) Show that the operations of addition of complex numbers and multiplication by a real
       number $a$ define a structure of a vector space [over $\bR$] on
       $\bC$.

 \item (3 points) Show that $\set{1,i}$ is a basis of $\bC$ as a real vector space.

 \item (3 points) Let $z=a+bi,w=c+di$ (with $a,b,c,d\in \bR$) be complex numbers. We define
the product $z\cdot w$ of $z$ and $w$ by $z\cdot w=(ac-bd)+(ad+bc)i$.
 Show that for any $z\in \mathbb C$ the map
$A(z)\from \bC \to \bC$ given by $w\mapsto z\cdot w$ is linear.

 \item (3 points) Write the matrix of $A(z)$ in respect to the basis $\set{1,i}$ of $\bC$.
\end{enumerate}

\item (10 points) Let $A,B$ be two $2\times 2$ matricies such that $AB=0$. Does this
equality imply that $BA=0$?

\item (10 points)
\begin{enumerate}
 \item (5 points) Let $V$ be a vector space of dimension $d$.
  Prove that one can not find a strictly increasing sequence $L_1
  \subsetneq L_2 \subsetneq \dots \subsetneq L_r$ for $r>d+1$, where
  the $L_i$ are subspaces.

 \item (5 points) Let $A$ be a $2\times 2$ matrix such that $A^5 =0$. Prove that $A^2 =0$
\end{enumerate}
\end{enumerate}
\end{document}
