1. Let P3 be the space of polynomials of degree at most 3, i.e., P3 = {po+ p1x + p2x² + p3x% : po, P1, P2, P3 € R}. Let T : P3 → P3 be the mapping defined by Tf (x) = f"(x) – 4f'(x) +f(x). Show A = {1, 1+x, (1+x)², (1+x)³} is a basis for P3. Find the matrix representation of T with respect to the ordered basis A = (1, 1+x, (1+x)², (1+x)³). %3D %3D

1. Let P3 be the space of polynomials of degree at most 3, i.e., P3 = {po+ p1x + p2x² + p3x% : po, P1, P2, P3 € R}. Let T : P3 → P3 be the mapping defined by Tf (x) = f"(x) – 4f'(x) +f(x). Show A = {1, 1+x, (1+x)², (1+x)³} is a basis for P3. Find the matrix representation of T with respect to the ordered basis A = (1, 1+x, (1+x)², (1+x)³). %3D %3D

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 6E

See similar textbooks

Similar questions

Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
3. For each of the following mappings, write out and for the given and, where.
Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
In Exercises 1-12, determine whether T is a linear transformation. T:FF defined by T(f)=f(x2)
Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
Let T be a linear transformation from P2 into P2 such that T(1)=x,T(x)=1+xandT(x2)=1+x+x2. Find T(26x+x2).
Prove that the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.
For a continuous transformation ƒ : X → X, prove that f is topological-transitive if and only if for any two nonempty open sets U, V Ç Xthere exists n E N such that f"(U) V ‡ Ø.
Find a linear mapping F: R3→R3 whose kernel is spanned by u1=( 2, 2, 1)and u2=(1, 0, 2)