center of G is defined to be Z(G) = {a E G | ag = ga for all ge G}. (a) Show that if X C G is a generating set for G, then Z(G) = {a E G | ax = xa for all x € X}. (b) Find the centers of Sn, Dn, and GLn. (Hint for GLn: There is a theorem from linear algebra that every invertible matrix can be written as a product of ele- mentary matrices. In other words, GLn is generated by elementary matrices.)

Q: Prove part 2 of Theorem 2.1. Part 2 of Theorem 2.1 states “For every a ∈ G, ø(a)-1 = ø(a-1).

A: Let G,* be a group and ∅:G→G be a homomorphism.  To prove that for every a∈G, ∅a-1=∅a-1 Since G is a…

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Q: Consider the following polynomials in P • P4(x)=x+1; Ps(x)=²+1; • Pu(x)=2²³ +1. P₁(x) = x³ + x² +1;…

A: The given polynomials are p1x=x3+x2+1p2x=x3+xp3x=x2+xp4x=x+1p5x=x2+1p6x=x3+1 (a) We have to find a…

Q: (b) Since (AB)*AB = B*A*AB = B*B = I and AB(AB)* = ABB* A* = AA* = I, we see that AB is unitary.…

A: Since A and B are unitary operators, so <A(x), A(x)> = <x, x> ........... (1) <B(x),…

Q: Which are invertible

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Q: Suppose T: R4→P3 is an isomorphism whose action on a basis for R4 is as follows: fil-attiva frontana…

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Q: Find the kernel and range of each of the followinglinear operators on P3: L (p(x)) = p(0)x + p(1)

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Q: Consider the matrix A = substituting z = A: (8) 0 Define an "evaluation" homomorphism: R[x] - → M₂…

A: Note: We will be solving part (c) and (d) only as asked.The given matrix is .Consider the…

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Q: Find the d. c. of T from definition.

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Q: 2πi 4. Let w = e, and let E = Q(w). (a) Verify that [E: Q] = 10. (b) Determine |Gal(E/Q)], the…

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Q: 11. Let f (x) E Z[x] with deg (f(x)) > 1. Let f'(x) be the polynomial in Zp,p € Z, obtained from…

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Q: Find the representation of I with respect to (a) S; (b) T ; (c) S and T ; (d) T and S.

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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Q: 3. Suppose V is finite-dimensional and S, T are linear operators on V. Prove that ST is invertible…

A: Proof are in given below Explanation:Step 1:

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Q: (1) Show that T: R² → P₁ given by T[B] = a + (a + b)x is an isomorphism.

A: A linear map T is said to be isomorphism if it is one to one and onto.

Q: Prove (b) of Theorem 6.15.Theorem 6.15. Let V be an inner product space, and let T be a normal…

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Q: Let T : R? → R² and T2 : R? → R² be linear transformations defined as follows. n (:)-L) -3x1 2x1 +…

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Q: III Linear Transformations 6. Label each of the following statements as true or false. If true,…

A: Note : '' As per guidelines we will solve first three subparts. If you want any specific subpart to…

Q: Find the transform in Z of the following exercise:

A: Given,          z2-7z(z-1)2(z-3)

Q: Q2. (a) Prove that if ), and >. are segma algebras over X, then 2n2, is a sigma algebra over X. (b)…

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Q: 6. Let T : Rª –→ Rª be given by T((x1,x2, x3, x4)) = (x1 – 2, x2 – x1,0, x3 + x4). It is known that…

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Q: Find the kernel for the transformation T(x) = Ax if ? = [1  2   0   1 −1]       |2   1   3   1   0|…

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Q: (This is closely related to "Taylor polynomials" which appear at the end of 1st year calculus.)

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Q: dy y X, dx

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Q: Find the transform in Z of the following exercise: z² (z-1)²(z²z+1)

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