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Elements Of Modern Algebra
- 12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.arrow_forward14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.arrow_forwardLet R be any ring. If f(X) = ao + α₁X + a₂X² + . = ao + a₁X + a₂X² + ... + a₂X¹ € R[X], define the anX" (formal) derivative of f by f'(X) = a₁ + 2a₂X+nan X-1. Prove that for any polynomials f(X), g(X) € R[X], (ƒ(X) + g(X))' = f'(X) + g'(X) (f(X)g(X))' = f(X)g'(X) + f'(X)g(X)arrow_forward
- 1. If f is the map that sends each complex number y z = x + yi → [-y Show that f(z,z2) = f (z,)f(z2),Vz1, Z2 E Carrow_forward11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) + f(y) and f(xy) = f(y)f(x) x, y € R. Let f, g be two antihomomorphisms of a ring R into R. Prove that fg: R R is a homomorphism.arrow_forward2. Consider the following functions. Are these ring homomorphisms? If yes, prove it. If no, provide a counterexample. a) f: ZZ given by f(x) = 3x. b) g: R R given by g(x) = x² - c) h: Z→ M(Z) given by h(a) = [ a 8arrow_forward
- Prove that the dual of l' is isometric to 10⁰.arrow_forwardLet R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a homomorphismarrow_forwardSuppose the first few Fourier coefficients of some function f in C[0,2n] are an, a1, a2, and b,, b,, b3. Which of the following trigonometric polynomials is closer to f? Defend your answer. ao g(t) = + a, cost+a, cos 2t + b, sint 2 h(t) = + a, cost+a, cos 2t + b, sint+ b, sin 2t 2 ..... Choose the correct answer below. O A. The function h(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and h(t) is the orthogonal projection of f onto that space. O B. The functions g(t) and h(t) are not members of the same space, and they are both orthogonal projections. It cannot be determined which function is closer to f. OC. The function g(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and g(t) is an orthogonal projection that requires fewer terms to approximate f. O D. The function h(t) is a member of a higher order space than g(t) so it must be a closer…arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning