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Linear Algebra
MAS4105
6137
ClassD
Prof. JLF King
16Nov2011
OYOP: For the 2 Essays:
Write your grammatical
English
sentences
on every
third
line, so that I can
easily
write between the lines.
For a square matrix
M
, the
“
minimum polyno
mial
of
M
”
[
minpoly
] is the
smallest degree
monic
polynomial Υ
M
() st. Υ
M
(
M
) is the zeromatrix. It is
always a polynomialdivisor of the charpoly of
M
.
D1:
Use
h·
,
·i
for an innerproduct on
R
vectorspace
V
.
x1
State
the
CauchySchwarz Inequality Thm
, carefully
stating the IFFcondition for equality.
x2
Prove
the
CS Inequality Thm
, using the axioms for innerproduct.
x3
Triangle Inequality Thm
:
For all
u
,
w
∈
V
, we have
that
k
u
+
w
k
6
k
u
k
+
k
w
k
.
Prove
the
TriangleInequality Thm
, using the
CS thm
.
D2:
Suppose
S
and
T
are 3
×
3 matrices, with
S
invertible
.
Let
f
() be the charpoly of product
ST
. Let
g
() be the
charpoly of
TS
.
Carefully prove
that
f
=
g
.
D3:
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This note was uploaded on 01/26/2012 for the course MAS 4105 taught by Professor Rudyak during the Fall '09 term at University of Florida.
 Fall '09
 RUDYAK

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