The problem is from Linear Algebra Done Right by Axler. Could you teach me how to do this in detail? Suppose T = L(V) and v € V.(a) Prove that there exists a unique monic polynomial p of smallestdegree such that p(T)v = 0.Prove that p divides the minimal polynomial of T.(b)

Suppose T∈Lv and v∈V. We know that a monic polynomial is a single variable polynomial in which leading coefficient is 1. To prove the given statements. (a) Now to prove that there exists a unique monic polynomial p of smallest degree such that pTv=0. Assume that there exist another monic polynomial qt of smallest degree. We know that p divides minimal polynomial of T. ⇒pt|gt ⇒gt=cpt Since gt and pt have same degree and c≠0 then c must be 1. ⇒gt=pt Thus there exists a unique monic polynomial p of smallest degree such that pTv=0. (b) First we will show that p divides minimal polynomial of T. Let us consider gt is minimal polynomial. Also consider pt is any polynomial. By division algorithm there exists polynomials say qt and rt such that gt=ptqt+rt where rt=0 or degrt<deggt. If degrt<degpt then rt must be minimal polynomial but this is not true as we assume gt is minimal polynomial. ⇒rt=0 ⇒gt=ptqt ⇒pt divides gt. Thus p divides minimal polynomial of T.

Answered: Suppose TEL(V) and v € V. (a) (b) Prove…

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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The problem is from Linear Algebra Done Right by Axler. Could you teach me how to do this in detail?

Suppose T = L(V) and v € V.
(a) Prove that there exists a unique monic polynomial p of smallest
degree such that p(T)v = 0.
Prove that p divides the minimal polynomial of T.
(b)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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