Use the result of Exercise 41 to show thatf1−α,ν1,ν2 = 1fα,ν2,ν1

Q: Let {u, v} be an orthonormal set in R" and let y = v13 u + 6v. Then, y|| is

A: Here, given that u,v are orthonormal. That is: u·u=1, v·v=1, u·v=v·u=0

Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express zx andzy in terms of fu , fy , and the…

A: Given, Let z=f(u,Y) u=ax+byY=ax-by

Q: The linearization at a = 0 to v8 + 5x is A+ Bx. Compute A and B. A = B = ||

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Q: Prove that Cov (x, y) = E (xy) – E (x) E (y).

A: Cov(x,y) is covariance of x and y. E(x) is expected value of x E(y) is expected value of y.

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Q: Prove that Vn > 0: 2º + 21 + .…+ 2" = 2n+1 – 1. ... an+1_1 Prove that Vn > 0: aº + a² + .….+ a" =…

A: Here, the objective is to prove the following: 1) 20+21+22+......+2n=2n+1-1   ; n≥02)…

Q: Evaluate (2z + z – i) dz J|z-il=1

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Q: Q2= A) for E= 2xĭ-3Y3+K %3D prove That divc curl È)=0 %3D

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Q: Exercise 2.11. Prove that n = for all n E Z. Exercise 212 Determnine whether or not the given…

A: We proof given statement by induction

Q: Use Cramer's rule to solve for y without solving for the un- knowns x, z, and w. 4x + y+ z + w= 6 .…

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Q: Give the condition which ensure that Je-| < 1 where z E C.

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Q: The linearization at a = 0 to /3 + 9x is A + Bx. Compute A and B. A = B =

A: Given: f(x)=3+9x

Q: a) Show that fi (x) = cos x, f2(x) = cos 2x are orthogonal on [-7, n].

A: As per our guidelines we can answer only 1 question so kindly repost the remaining questions…

Q: In Exercises 23–30, find Tn centered at x = a for all n. 26. f (x) = e", a =-2

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Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

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Q: Exercise 5.10. Suppose that Z = {Zn}n EZ.. Show that Z is a martingale.

A: Martingale: Let {Fn} be an increasing sequence of algebras in probability space referred as…

Q: Let X₁, X₂,..., X be a r.s. from N(,0) and Y₁, Y₂...., be another independent r.s. from m…

A: Solution

Q: If u = x3 -3xy and v= 3x2y-y3, prove that Pu/Əx2 + Pu/əy2 = Pv/ax2 + Pv/@y²

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Q: Let y1 = 9cos(2x) and y2 = 2cos2 (x) – 2sin? (x). Calculate the Wronskian, W(y1, Y2). Give your…

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Q: verify that (A + B)T = AT + BT

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Q: Exercise 4 Show that for cach k EN it holds that kn-k < n! S n", Vn E N, n2 k. Deduce from this that…

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Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express z x and zy in terms of fu , f y, and the…

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Q: The linearization at a = 0 to v4 + 4x is A + Bx. Compute A and B. A = B = %3D II

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Q: I. Solve the problem by using Jordan canonical form Exercise 3.2.18 Use Cramer's rule to find the…

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: prove that erf (x) dx =o

A: To prove: ∫−aaerfxdx=0

Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: Find the linear approximation of fcn)cn" In n around n=e Using this equation, y = f(a)+f'ca)(n-a)

A: Applying linear approximation we have

Q: Use Cramer's rule to solve for y without solving for the un- knowns x, z, and w. 4x + y + z+ w= 6 3x…

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Q: Exercise 3.68. Let Z~ N(0, 1) and X ~ N(u, o²). (a) Calculate E[Z*J. (b) Calculate E[X*].

A: Given Z ~ N(0,1) and X ~ N(µ,σ2) a) If Z follows Normal distribution, then Z2 follows Chi-square…

Q: Let x, y ∈ R. Suppose x2 + y2 = 0. Prove that x = 0 and y = 0.

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Q: Exercise: 1- Show that exp (1 z) = exp(1 z) z =n n,n E Z. 2- Find all values of z such that a) ez =…

A: Note: There are multiple sub questions. No specific question has been asked. So, We will solve the…

Q: Use Cramer's rule to solve for y without solving for x, z, and w. 5 x + y + z + W = -7 5x + 3y - z +…

A: Here we have to use cramer rule for solving the system of linear equations

Q: Let F I2). Evaluate JJaw as z² + y? 0

A: In the given question, the concept of the Divergence Theorem is applied. Divergence Theorem Assume S…

Q: LZHP - DTNV. Complete the statement below. ZZ = _2_

A: Consider the following condition: LZHP~DTNV Hence, ∠L≅∠D∠Z≅∠T∠H≅∠N∠P≅∠V

Q: Calculate A and B so that t B %3D 1+1 (1-z²)

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Q: Find the positive integer A such that t²x" +tx' + t²x = λx has a nontrivial solution satisfying x(0)…

A: Hint: Substitute t=ey , t2x''=D(D-1), tx'=D-1in the given equation and calculate to obtain the…

Q: Show that the functions x and x2 are orthogonal in P5 with inner product defined by (5), where xi =…

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Q: . Let Show that, Hence show that •2T F₁(x) = 2/1/² 0 Fn(2) = eix cos ∞ N=-∞ eix cose eine de in Jn…

A: As per the question we are given the sequence of functions Fn(x) in form of a Fourier integral. And…

Q: Let u(x) be a solution of u" – q(x)u = 0 with q(x) > 0 such that u(0) and u'(0) are positive. Show…

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Q: dv Prove that 4u/2n

A: We use Gamma function to prove the given result.

Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: Exercise 3.68. Let Z~ N(0, 1) and X N(u,o2). (a) Calculate E[Z*). (b) Calculate E[X*].

A: We have given that, Z~N( 0, 1 )    and  X~N( μ, σ² )   We know that in general, E(Y2n) = μ2n =…

Q: Exercise i) Calculate gcd(306,657) ii) Find all x, y E Z such that gcd(24,138) = 24x + 138y

A: I have find gcd by basic rule and then used the division algorithm

Q: Show that e", sin x, cos a are linearly independent using Wronskian Method.

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Q: 2. Let X ~ U[0, 1] and Yn = X. Prove that Y, 4x.

A: Given that, X~U0,1 Yn=n-1nX

Q: Reduce Z= ( 1+i) (1+i) / (1-í), to the form a + ib -1-i .a 1-i .b 1+i .C -1+i .d others .e

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

A: ANSWER: