Let p: M M' be a homomorphism of R-modules and let N be an R-submodule ofM. We define the image of N under y to beP(N) := {y(u) : u e N}.Show that(a) p(N) is an R-submodule of M';(b) if N = (u1,..., Um) R, then o(N) = (p(4),...,p(um))R(c) if N admits a basis {u1,..., Um} and if p is injective, then {9(u1),...,(um)} is abasis of (N).%3D

I have used the submodule criterion and definitions of generator's of module and basis of module

Answered: Let p: M M' be a homomorphism of…

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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Let X be a lattice
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Exercise 4.2.5. Let q: Z6 → Z2 × Z3 be the map q([k]6) = ([k]2, [k]3). (1) Show p is well-defined. (2) Show p is a homomorphism. (3) Show q is injective. Why is this enough to conclude q is an isomorphism?
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Let B be an invertible matrix nx n and M₁ (R) be the space of all nxn metrices. Prove that the map f: M₂ (R) → M₂ (R) defined by f(A) = AB is a bijective endomorphism.
2. Recall that in class we proved the following theorem: Theorem. Let T: VW be a linear transformation between finite-dimensional inner product spaces, and let T*: W → V be the adjoint of T. Then N(T) and R(T*) are orthogonal complementary pair in V, and R(T) and N(T*) are orthogonal complementary pair in W. Use this theorem to derive the following corollary: Corollary. Let A E Mmxn (R). Then the row space of A and the solution space of Ax = 0 are orthogonal complementary pair in R", and the column space of A and the solution space of Aty = 0 are orthogonal complementary pair in Rm.
Let T : V → W be a linear transformation. Let B = {bi, b2 ·.. , bn} be a spanning set of V. (a) Prove or disprove: The set {T(b1),T(52), ··… ,T(b,)} is a spanning set for Im(T). (b) Prove or disprove: dim(imT) < n. (c) Prove or disprove: dim(V) = n.
4. Let w = e 11, and let E = Q(w). (a) Verify that [E:Q] 10. (b) Determine |Gal(E/Q)|, the number of automorphisms of E that fix Q. (c) Find the order of y(w) w? in Gal(E/Q). (d) How many subfields K does E have (such that QC K C E)? Hint 1 The corollary to Eisenstein's theorem might help. Hint 2: Example 6, page 535 may help.
Use First Isomorphism Theorem to prove that R/ZE S'.

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