Let V be a finite dimensional vector space over a field F and let T be a linear operator on V. Then there exists a vector in V such that a; m (x) = m+ (x).
Let V be a finite dimensional vector space over a field F and let T be a linear operator on V. Then there exists a vector in V such that a; m (x) = m+ (x).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.6: Algebraic Extensions Of A Field
Problem 14E
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do both
![Let V be a finite
dimensional vector space over a field
F and let T be a linear operator on V. Then there exists
a vector a; in V such that
'j
m
a¡
(x) = m+ (x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb1779f0a-3fd2-4949-87ba-9925e9c46523%2F298fea6a-29ab-45a5-8289-4caa8fcb272f%2F5st4und_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let V be a finite
dimensional vector space over a field
F and let T be a linear operator on V. Then there exists
a vector a; in V such that
'j
m
a¡
(x) = m+ (x).
![If A E F is a characteristic root of T then A
is a root of the minimal polynomial of T.
In particular, T only has a finite number of
characteristic roots in F.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb1779f0a-3fd2-4949-87ba-9925e9c46523%2F298fea6a-29ab-45a5-8289-4caa8fcb272f%2Fh1l6ffd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If A E F is a characteristic root of T then A
is a root of the minimal polynomial of T.
In particular, T only has a finite number of
characteristic roots in F.
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