1. Explain simplify 599)..0. Let F be a field, let f(x) = co+c++ cd-12-1+xd be a monic polynomial in F[r], andlet T be the operator on Fd with matrix (with respect to the standard basis)0the companion matrix to f(x).10C₁ =0(a) Show that the minimal polynomial for T is f(x).(b) Show that XT(x)=(-1)df(x).-CO.. 00-cd-201 -Cd-1

###(a)To show that the minimal polynomial of T is f(x) , we need to prove two things:1. f(T) = 0…

Answered: 599). . 0. Let F be a field, let f(x) =…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

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Similar questions

Find a basis for R2 that includes the vector (2,2).
11. Show that defined by is not a homomorphism.
Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.
Prove that every ordered integral domain has characteristic zero.
Let V denote the space of polynomials in x of degree ≤ 3 over a field K. Letv1 =1+x+x2 +x3, v2 =1+2x+3x2, v3 =2+6x, v4 =6. Are the vectors v1, v2, v3, v4 linearly independent over each of K and explain why. (i) K=R; (ii) K = F3, the field of integers mod 3; (iii) K = F5, the field of integers mod 5.
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
Let f(x) = x² + 1 € Z3[x) and let R= Z3[x]/I, where I = (f(x)). (a) Show that R is a field with 9 elements. (b) Denote by 0 := 0 + I, 1 := 1+ I, and a := x + I. Write the other 6 elements of R in terms of a and determine the multiplicative inverse of each nonzero element. (c) Prove that R Z3[i].
5. Let F be a field and let V=Fnxn. Let C be a fixed element of V and let To be the linear operator on V defined by Show that det Tc = 0. Tc (A)= AC-CA.
Let R[x] be a linear space with all polynomials p(x) = ax² + bx + c (a, b, c € R) whose degree is no more than 2. Let L be the operator on R[x]3 defined by 4 2 L(ax² + bx + c) = (a + 3b +6c)x² + (-=a 3b4c)x+ ²a + 3 3 (1) Find the matrix A representing L with respect to basis [1,2x, 3x²]; a + b + c) (2) Find the matrix B representing L with respect to basis [1-2x+3x2²,-1+2x, −1+3x²]; (3) Is the matrix A diagonalizable? Why?
Let P3 be the vector space of all polynomials of degree 3 or less in the variable z. Let = 2+x+x², 2+x+x², 2+x², = 11 + 3x + 6x² choose PI(T) P2(x) P3(x) = P4(x) and let C = {p1(x), p2(x), P3(x), P4(x)}. a. Use coordinate representations with respect to the basis B = {1, 2, ², ³} to determine whether the set C forms a basis for P.. = c. The dimension of span(C) is b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. {}

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