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\title
Math 262 -- Linear Algebra\vskip 10pt
Spring 2000 (Final Exam )
\endtitle





\vskip20pt

1) Prove the \it {parallelogram law} \rm on an inner product space 
$V$; that is, show that for all vectors $x$ and $y$ in $V$
$$||x+y||^2+||x-y||^2=2||x||^2+2||y||^2.$$
What does this equation say about parallelograms in $\Bbb R^2$?

\vskip20pt

2) Consider the space $M_{n\times n} (\Bbb R)$ of all $n\times n$ 
real matrices endowed with the inner product $\langle A,B\rangle=$ 
trace of $B^{\ast}A$. Consider
the subset $W$ of $M_{n\times n} (\Bbb R)$ consisting of those 
matrices of zero trace. Show that $W$ is a subspace of dimension 
$n^2-1$ and find its orthogonal
complement.

\vskip20pt

3) Give examples of:
\vskip10pt
a) a $3 \times 3$ nilpotent matrix of rank 2.
\vskip10pt
b) A $9 \times 9$ matrix in Jordan form with three Jordan blocks and 
a double eigenvalue.


\vskip20pt

4) Let $N$ be a nilpotent $n \times n$ matrix. Show that the matrix 
$I-N$ is invertible (you may either use determinants or look at the 
power series
expansion about $x=0$ of the function $(1-x)^{-1}$ for inspiration).

\vskip20pt

5) let $V$ be a 14-dimensional vector space, $W_1$, $W_2$ subspaces 
of dimensions 8 and 10, repectively. What are the possibilities for 
the dimension of
$W_1 \cap W_2$?

\vskip20pt

6) a) Give the definition of the minimal polynomial associated to an operator.
b) Formulate and prove a criterion for an operator on a finite -- 
dimensional vector space to be triangulable.  


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