1) Let ACP(R), let f: R → R, and set A' = {0 CR : f-¹(O) E A}. Prove that A' is a o-algebra. Assume A is a 6-algebra !!!

Q: Suppose L: R² R³ is defined by L the given bases. [L(u)]s, = [L(u)]B₂ = ~Q[*^~-{]}]}----(HH:)…

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Q: Show that the mapping Te data by pla + b/2)=a- bv2, a,beQ is an automorphism of the body Q(/2). Show…

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Q: Let R = - b ,be Z}and let ø : R→Z be a mapping defined by : b = a-b. a a

A: We will find out the required value.

Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~V* as…

A: To show that for any tow finite-dimensional L-representation V and W of L HomkV, W≃V*⊗W as L-module

Q: The splitting field of f=(x^2-2)(x^2+1) over ℚ is ℚ(sqrt(2),i) because its roots are…

A: Please find the detailed solution in next step.

Q: Let phi / Q * (sqrt(2)) -> phi(sqrt(2)) be the map given t phi(a+b sqrt 2 )=a-b sqrt 2 . Show that…

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Q: Let K and K' be algebraic closures of a field F, then there exists an ismorphism o: K→K' of Konto K'…

A: Let K and K' be algebraic closures of a field F, then we have to show that there exists an…

Q: One of the following is not integral domain a) (Z, +7,7) b) (R, +,) c) (C,+,) d) (P(X),A,0) a) O d)…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Give the splitting field of x4 - 1 over 1 over Z2, Z3 and Z5.

A: Given: Objective: To give the splitting field of  over  and .

Q: Let F(a) be the field described in Exercise 8. Show that a2 and a2 + a are zeros of x3 + x + 1.

A: Given: Fa is a field. To show: a2 and a2+a are the zeros of x3+x+1.

Q: Let F = {0, 1} be the field with two elements, and let · denote the standard dot product in F100,…

A: Given: F=0,1 is a field with two elements. "·" denotes the standard dot product in F100.…

Q: I show that T:ℂ2 -> ℂ2, defined by T((w,z)) = (iw, z-w) is 1-1 and onto

A: Introduction: A function using several numbers as input or output is referred to as a multivariable…

Q: Let f R2 R² be an isomorphism where : → ƒ ( [¹³]) = [()] and ƒ ([×4]) = }}] Find(G)

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Q: Consider the linear operator L: BC[0, ] → BC[0, 7] defined by Си — и" - 2и is unbounded in the sup…

A: Consider the linear operator L:BC0,π→BC0,π defined by: Lu=u''-2u. We know that the set BC0,π is…

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: Prove that the following does not define an inner product R?: ((a, b), (c, d)) = ac – bd. on

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Q: let (R₂ *) be a mathematical system * is defined as: FabER where axb=atbab prove that: □ is…

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Q: Q2/(A): Let L.N are submodules of M and n:M - M/N be the natural R-homomophism. Is n(L) submodule of…

A: Given that,                 L and N are submodules of M .  Also, π : M → MN is a natural R-…

Q: (3x)i + (3y)j + (3z)k F = x3 + y3 + z3

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Q: polynomial in Z_2[x] (where Z_2 is the field containing two elements 0 and 1)? a) x^2 b) x^2+x c)…

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Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

A: We have the function F:ℝn→ℝn that preserves the norm · This implies that this function satisfies all…

Q: 11) If p 2 1, A,B e R".Then, |A+ Bll, s ||A, + ||B|, is called the Minkowski inequality a) True (b)…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: In the frieze group F7, show that zxz = x-1.

A:

Q: A) Now, let R E L(V) be a self-adjoint SE L(V) a normal operator, and U E L(V) an operator that is…

A: As per Bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: Theorem: if I is a Positive implicative Bck-algebra Then (Lx),,Lo) is a BCK-algebra

A: Consider the given theorem, '' if X is positive implicative Bck-algebra then Lx,⊙,L0 is a…

Q: True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).

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Q: Let A€BL(H) be a compact operator, then show that A*A is a compact

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Q: Use membership table to prove (AunionB) – C = (A – C) union (B - C).

A: We need to prove the following statement using membership table. A∪B-C=A-C∪B-C

Q: Define complemented lattice. Let (L, *,Ð, ', 0,1) be a Boolean algebra. Then show the following:…

A: Given: (L,*,⊕,',0,1) is a Boolean algebra.We need to define complemented lattice and show that…

Q: Let V be an n-dimensional vector space over a field F, where n ≥ 1. Prove that L(V, F) is isomorphic…

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Q: ve that the Bessel functionm generating funchion ie too

A: The aim is to show that∑n=-∞∞Jn(x)tn=ex2t-1t The approach for this proof is, Take,…

Q: Please find all automorphisms. Thanks.

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Q: In the frieze group F7, show that zxz = x-1.

A: As per the question we are given the frieze group F7 which consists to four group operations : Tx :…

Q: What de you. under stand by extenSion- field? IS there any relation of extension- field and vector…

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Q: Check whether or not if L/K and K/F are Galois extensions, then L/F is a Galois extension.

A: Suppose that EF and KF are extensions of a common base field F. Say that a field homomorphism φ: E→K…

Q: Can you help me explain why T is onto if Rank(T)=2 ?

A: