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…

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

Suppose TEL(V) and v € V. (a) (b) 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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 19E: Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in...

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Question

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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