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\title
Math 262 -- Linear Algebra\vskip 10pt
Spring 2000 (Exam 3)
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\vskip 20pt


\vskip30pt

1) Write an essay about the Jordan canonical form for operators on 
finite - dimensional complex vector spaces. Your discussion must 
include the primary
decomposition theorem, the decomposition of an operator in terms of 
diagonalizable and nilpotent operators, etc... You must explain 
clearly how these
elements fit together to yield the Jordan form.  Give at least two 
examples to illustrate the various possibilities.

\vskip30pt

2) Let $V$ be a finite - dimensional inner product space, $V^{\ast}$ 
its dual space. Prove that there is a $\underline {natural}$ 
isomorphism $\Lambda :V\rightarrow
V^{\ast}$ (give full details).

\vskip30pt

3) Denote by $O(3)$ the set of all orthogonal linear transformations 
$T: \Bbb R^3\to \Bbb R^3$ such that $\det T=1$. Show that $O(3)$ is a 
group under the
usual notion of product of operators. For a fixed $p\in \Bbb R^3$, 
what is the subset of $\Bbb R^3$ defined by $\{Tp \;| \;T\in O(3) \}$?

\vskip30pt

4) Apply the Gram - Schmidt orthogonalization process to your 
favorite non- orthogonal basis of $\Bbb R^4$.

\vskip30pt

5) Using the spectral theorem, explore the possibility of defining 
$f(A)$, where $f:\Bbb R\to \Bbb R$ is any function and $A$ is any 
symmetric $n\times n$ real
matrix. 

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