Let T : V → U be a homomorphism, then Ker T = { 0 } iff T is one-one

Q: (B): Let a:M → N, B: N →M are R-homomorphism such that a oB = In then M = Ker(a) + Im(B) and Ker(a)…

A: The problem is related with R homomorphisms. Given: α :M → N , β :N →M are R homomorphism, such that…

Q: Let f: R R given by f(z, 9) = (x+2y, z- y). Of is 1-1 of is not 1-1 of is not onto of is onto

A: Given that fx,y=x+2y,x-y

Q: Compute Ker(ø) for the homomorphism ø: Z→Z, such that (1)=9 6Z 8Z O None of them 5Z 7Z

A: Given∅(1) = 9 then ∅(2) = 2∅(1)= 2(9)= 18= 3 mod 15= 3 Hence ∅(2) = 3 Also ∅(3) = 3∅(1)= 3(9)= 27=…

Q: 5. Let a E G be fixed. Define y: G → G by p(x) = axa-1 for all x E G. Prove that p is a…

A: To find - Let a∈G be fixed. Define ψ:G→G by ψx=axa-1 for all x∈G. Prove that ψ is a homomorphism.

Q: 4. Let y: R→ S be a ring homomorphism. Prove that ': R[X] → S[X] given by soʻ (ao + a1X + ..a,X") =…

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Q: Suppose A is a normed algebra and define the map MB : A → A by MB(A) = BA. Then Mg is continuous on…

A: A is normed algebra Mb: A→AMb(A)=BA

Q: 16. Let P : R" → R" and Q : R" → R™ be linear mappings. (a) Prove that S: R" → R™ defined by S(x) =…

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Q: If E is an extension of F and f (x) e F[x] and if o is an automorphism of E leaving every element of…

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Q: 3 Let f: (A, SA)(8, 5,) be an order isomorphism and, Sa={xEA : x<ga}+ and Spa) -ly e8: f(5.)=Spa)…

A: Given: f: (A,⩽A) → (B,⩽B) is an order isomorphism Sa={x∈A: x&lt;Aa}≠ϕ Sf(a)={y∈B: y&lt;Bf(a)} To…

Q: Let R=Z and R'= set of all even integers. Then %3D (R', +, *) is a ring, where a* b= ab V a, be R'.…

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Q: Which of the following is not an automorphism? * Ø: (Z, +) → (Z, +) with Ø(x) = 3x %3D This Option…

A: Use the definition of Automorphism to solve this problem.

Q: 23. Let 0:→ be defined by a b = a+d (a,b,c,de R ) a) Prove that 0 is a homomorphism b) Determine…

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Q: If E is an extension of F and f (x) e F[x] and if o is an automorphism of E leaving every element of…

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Q: f:A→Z is not onto and f:A M is onto.

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Q: If E is an extension of F and f (x) e F [x] and if o is an automorphism of E leaving every element…

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Q: 6. Let f : R? →R defined by f(1, z2)) = 11 – 12. Then determine if is (a) Lipschitz continuous(if it…

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Q: Show that the mapping :S3 S3 given by p(x) = xis not a homomorphism.

A: Solution is below.

Q: Functional theory and functional analysis

A: Consider the provided question, Let A be an invertible operator acting on space X.

Q: Consider p: R -R under addition, defined by (x) = x². %3D a) o is a homomorphism b) o is not a…

A: A mapping f: (G,+)→(G*,·) is said to be homomorphism if it satisfies following condition:…

Q: Let p: GL(2, R) → R' be a homomorphism map defined by p(A) = det(A), then %3D a) Ker o = GL(2, R) b)…

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Q: (b) If P is the projection on ce M of a Hilbert space H th riant under an operator T

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Q: (d) Is T an isomorphism?

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Q: Prove that R* ⨁ R* is not isomorphic to C*.

A: First, we count the elements of finite order; R has two elements of finite order that is (1, -1) C…

Q: E. Let o : R+ → R* under multiplication be given by o(x) = |x|. 1. Show that is a homomorphism. 2.…

A: ϕ: R+→R+ϕ(x)=x

Q: a Let S= { : a,b e R}. Show that : C→ S defined by %3D a b is a ring homomorphism. a $(a + bi) = -b…

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Q: Let n e N, q E Q and let E be the splitting field of r" F:= Q(e). Show that Gal(E/F) is abelian. q…

A: Let n∈ℕ, q∈ℚ and E be the splitting field of xn-q over F:=ℚe2πin To prove that GalE/F is abelian.…

Q: 9. Let F : R" → R". Prove that for each a the mapping h→ dF[a; h] is linear on R".

A: Given: F:ℝn→ℝm   To prove: For each a, the mapping h↦dFa;h is linear on ℝn

Q: Determine whether V and W are isomorphic. If they are, give an explicit isomorphism T: V---&gt;W   V…

A: Given, V = C, W = R2, and a map T : V ---&gt; W. we have to show that whether  V and W are…

Q: If E is an extension of F and f(x) belong to F(x) and if phi is an automorphism of E leaving every…

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Q: Recall that Q8 =. Prove that the map o from Qg to GL2(C) defined on generators by P(i) = |V-1 And…

A: A group homomorphism Φ : G→H is a map such that for any g1, g2 ∈G, Φg1g2=Φg1Φg2. The kernel of a…

Q: Let A = {" :a, b e Z} and let P: A → Z such that P(z) = a – b for every z e A. Prove that O is a…

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Q: Prove that T is a lincar transformation.

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Q: Exercise 8.12. Let : Z Z ⇒ Z via (a, b) → a + b. (1) Prove that is a homomorphism. (2) Find kero.…

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Q: Let f: MM' be an R-module homomorphism, then

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Q: 6. Let f: R? + R defined by f(r1, 12)) = #1 – a2. Then determine if is (a) Lipschitz continuous(if…

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Q: Prove that the mapping x --&gt; x6 from C* to C* is a homomorphism.What is the kernel?

A: Given: the mapping x --&gt; x6 from C* to C* is a homomorphism.

Q: 24. Let G = s C. Define 0: - by o( n) =i" a) Verify that o is a homomorphism b) Find Ker( o )

A: Apply Homorphism definition

Q: 1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{ACX: 2o ¢ A} is a topology on X. b)…

A: Topology means we have to prove 3 conditions then we can tell that space is Topology.

Q: Show that linear mapping  T : P2 (R) → R3defined by  T (ax2 + bx + c) = (a + b, a + c, b+ c), is an…

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Q: Let f: (G,.) (G') be homomorphisum, then ker(f) (e') iff is onto isomorphism O 1-1

A: No f is 1-1

Q: Suppose q: X →Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism. (Hint:…

A: Suppose q: X→Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism.

Q: Consider p: IR → R under addition, defined by p(x) = x2. a) o is a homomorphism b) o is not a…

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Q: Let R = {; la. b e z}and let p:R - Zbe defined by : 0( ) = a 1) is a ring a) Homomorphism. b)…

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: (2) Suppose UC R" and V C R" are open subsets, and we have C maps U V 4U such gof is the identity of…

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Q: Suppose q: X → Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism. (Hint:…

A: Given: q:X→Y is a quotient map and it is one-to-one. To show: q is homeomorphism.

Q: If E is an extension of F and f(x) belong to F[x] and if phi is an automorphism of E leaving every…

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Q: P: Sn → Z/2Z (0, if o E An (1, if o ¢ An P(0) Check to see if o is a homomorphism? Is p an…

A: It is given that φ:Sn→ℤ2ℤ φσ=0, if σ∈An1, if σ∉An Homomorphism:  φσ1·σ2=φσ1·φσ2 We have φσ1=0, if…

Q: prove or disprove : If F:(R,+) (R\{0},.), defined by f(x)= x +2 is an isomorphism.

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Q: Use First Isomorphism Theorem to prove that R/Z S'.

A: