1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) = (xz – yw, xw + yz). Then (1,2)-1 (a) (2,1) (b) (;,-) (e) -) (d) No Choice

Q: Theorem 2. Suppose u e F where F is an ordered field. Then u is positive f and only if u > 0.…

A: Note: According to bartleby we have to answer only first question please upload the question…

Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

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A: We have to first find all partial derivatives.

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A: An important result from advanced calculus, called the Implicit Function Theorem,states the…

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A: Givenf(x)=x2-2            and     x0 =0 newton method  xn+1=xn-f(xn)f'(xn)

Q: 2. Find all volves of tand u x(t,4)=0 and simultomebusly such that =0. 3. 5x-11* C.

A: Get the equations for:  fx=0fy=0 15x2-11y=0-11x+3y2=0 or 15x2=11y3y2=11x

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Q: 7. i) Show that Q(2'4,i)/Q is a Galois extension.

A: The given function is, E=ℚ214,ii,e, E=ℚ24,i The minimal polynomial of 24 is  x=24Taking power 4 both…

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A:

Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by f X,Y (x, y) = (x+y)/21…

A: The joint probability mass function of X and Y is  f(x,y)=(x+y)/21   , x=1,2,3 and y=1,2

Q: 4. Show that both functions x and sin(x) belong to inner product in this space. L² ([0, 1]) and find…

A: L20,1 is a vector space of all functions f:0,1→ℝ satisfying ∫01f(x)2dx<∞. And the inner product…

Q: (d) Z[ /2] is a field.

A: As you asked for option D.

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: Suppose that ℝ×ℝ, +, · is a field where (x, y) + (z, w) = (x+z, y+w) and x, y·z, w = xz-yw, xw+yz…

Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

A: Consider the given inequality 0≤x2-xy+y2  for all x and y in  ℤ Now,…

Q: Let X,X2, ,X, be a ranc *(X), = 0*(1– 0)1-x; %3D

A: from above distribution is binomial  f(x)θ=θx(1-θ)1-x  ;0<x<1 E(x)= nθvar(x)=nθ(1-θ)…

Q: 1. Show that sinz is unbounded on C and lyl <|sinz] <ebl z=x+yi € C. for

A: Given:         We have to show that sinz is unbounded in ℂ and y≤sinz≤ey   for z=x+yi ∈ℂ Function…

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A:

Q: 4: verify that wy = Wyx . ریاضيات المرحلة الثانية a w = In (2r + 3y) %3D

A:

Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

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A:

Q: Show that if AT Ay = y for all y in Rn , then ATA = In .

A: Given that     ATAy=y  ∀y∈Rn. To prove ATA=In

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A: To prove: Fx=8x4-8x2+1 is chaotic on -1,1. First plot graph of given function with -1,1 as follows:

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A: A 3×3 matrix a11a12a13a21a22a23a31a32a33.

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A:

Q: Evaluate the work done between the points P, (3,1,4) and P, (6,3,6) for the conservative field F =…

A:

Q: Compute ƒxz and ƒzz for ƒ(x, y, z) = xyz - x2 z + yz2.

A: Given that f(x,y,z)=xyz-x2z+yz2. We need to find : i)fxz. ii)fzz.

Q: Show that if and ßz are two Borel fields, then ß1 N B2 is also a Borel field.

A: Given that β1 and β2 are two Borel fields,we have to show that β1∩β2 is also Borel field

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A:

Q: Find Z-transform and ROC f x(n) = {1,2,−1,2,−1}

A: Given that: xn=1,2,-1,2,-1 The objective is to determine Z-transform and ROC. Let consider,…

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A:

Q: 1. Show that the definition (x1, x2) · (Y1, Y2) = x1Y2 is not an inner product on R?.

A:

Q: Find the Taylor expansion of 1/z² for points inside the circle |z – 2| < 2.

A:

Q: 1. Suppose 2 containing C. {0, 1} and C {{0}}. Enumerate N, the class of all a-fields %3D %3D

A:

Q: Is ZXZ cyclic? [Here Z means (Z, +).]

A:

Q: 5. Show that xZx + yZry = yezy if z= xy + y In xy

A:

Q: The polyromial X+X+1 is ireducilole aver Z2. Use it to Canstruct additlon and multiplcotion table of…

A: Let F8 be a finite field with 8 elements. Since the polynomial x3+x+1 is irreducible in ℤ2x,…

Q: In C[-1,1], with the standard inner product f(x)=ex and f(x)=e−x are orthogonal. true or false?

A: true please see explanation in step 2

Q: Prove that du Inlul + C where n= -1 u

A:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x +z, y + w) and (x, y). (z, w) =…

A:

Q: 5. Use Cauchy's Residue theorem to evaluate dz; Có lz-il =1 (2+1)

A:

Q: Prove that where e, > (1,0)"

A: Prove that     conve1, e2, -e1, -e2=x∈ℝ2 : x1+x2≤1 where e1=1, 0T , e2=0, 1T.

Q: 1. Suppose that (Q x Q, +,.) is a field where (x, y) + (z, w) = (x + z,y + w) and (x, y). (z, w) =…

A:

Q: 5. Show that for Chebyshev polynomials (Tn, Tm) = 0, n #m, π, EIN n=m= 0, n = m = 0.

A: We have to solve given problems;

Q: Let x, y, z ∈  ℕ. Suppose gcd(x, y) = 1. Prove that if x | yz, then x | z

A: In the given question we have to prove that if x|yz , then x|z and g.c.d(x,y)=1 and x,y,z∈ℕ. we will…

Q: 2. If A + 4, N, what is the Borel field generated by {A}.

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: 1. Suppose that (R × R, +, .) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: I have proved (1, 0) is the identity element and the checked by option one by one

Q: Show that if AT Ay = y for all y in Rn, then AT A = In.

A:

Q: 7. Suppose that f is the scalar field that turns each point (x1, X2, X3, ... , x100) in R100 into…

A: Given that f: ℝ100→ℝ, defined by, fx1, x2, ……,x100 =x1+ x2+⋯+x100 (a) Then, ∇fx1, x2, ……,x100 =∂x1+…