Suppose V is a finite-dimensional with dim V > 1 and T € L(V). Prove that {p(T)|p € F[x]} ‡ L(V). This proof can be done without assuming the T has an eigenvalue. (If the field of scalars is R or Q, then not every operator need have an eigenvalue.)
Suppose V is a finite-dimensional with dim V > 1 and T € L(V). Prove that {p(T)|p € F[x]} ‡ L(V). This proof can be done without assuming the T has an eigenvalue. (If the field of scalars is R or Q, then not every operator need have an eigenvalue.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![6) Suppose V is a finite-dimensional
with dim V > 1 and T € L(V). Prove that
{p(T) [p = F[x]} # L(V). This proof can be done without assuming the T has an
eigenvalue. (If the field of scalars is R or Q, then not every operator need have an
eigenvalue.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbdd7eaed-9b44-4003-81eb-7e8fab9c2687%2F602c6120-a446-4a3d-8722-84a8b0970e44%2Fu74jntv_processed.png&w=3840&q=75)
Transcribed Image Text:6) Suppose V is a finite-dimensional
with dim V > 1 and T € L(V). Prove that
{p(T) [p = F[x]} # L(V). This proof can be done without assuming the T has an
eigenvalue. (If the field of scalars is R or Q, then not every operator need have an
eigenvalue.)
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