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\title
Math 262 -- Linear Algebra\vskip 10pt
Spring 2000 (Exam 2)
\endtitle





\vskip 20pt

1) Let $T:V\to V$ be a linear operator, where $V$ is a 
finite-dimensional complex-vector space. Show that there exists a 
basis of $V$ with respect to which the
matrix of $T$ is upper triangular.

\vskip30pt

2) Show by means of an example that the conclusion in problem 1 may 
fail if $V$ is a real vector space.

\vskip30pt

3)  Consider the operator $T:\Bbb C^4\to \Bbb C^4$ given by 
$$T(z_1,z_2,z_3,z_4)=(z_1+iz_2+z_3+z_4,z_2,z_2+3z_4, z_2+4z_4).$$
\noindent
Is $T$ diagonalizable? What is the minimal polynomial of $T$?
\vskip10pt

\vskip30pt

4) Linear operators on a complex infinite -- dimensional vector space 
need not have eigenvalues. Check that the operator below is a 
suitable example: $V=$ space
of all continuous functions $f:[0,1]\to \Bbb C$, $T:V\to V$, $Tf=gf$, 
where $g$ is a fixed function in $V$ with the property that $g$ is 
non-constant on every
open subinterval of $[0,1]$.

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