[0, 1]. If f(1), g () e V then show that (f(t), g (t)) = [ƒ(t) g (f) dt defines an inner product on V (C).

Q: Let f € L'(0,1). Assume that f(t)t2"dt = 0 for all n f = 0 a.e. 1,2,3,.... Prove that %3D

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Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

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Q: Find the potential function f for the field F. F= (4x3y10z10, 10x4y9z10 10x4y1029) O fix, y. 2) =…

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Q: If the Wronskian of f and g is t cos t - sin t, and if u = f + 3g, v= fg, find the Wronskian of u…

A: It is provided that the Wronskian of f and g is tcost-sint. We need to determine the Wronskian of u,…

Q: Find the linearization L(x) at x= a. f(x) = x+ a = - 1

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Q: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: Give V = C([0, 1]) the inner product defined as follows: (f, g) :| f(t)g(t) dt for all f, g e V.…

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Q: Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function…

A: We have the given Cauchy problem is y'=a(x)tan-1y,  y(0)=1   …..(1) where a(x) is a continuous…

Q: Suppose that P1(t), p2(t), and p3(t) are continuous on R, and that y1, Y2, and y3 are solutions to…

A: Wronskian of the function

Q: Consider the following inner product: (S, 9) = /.2 f(x)g(x) VI dx. Show that the set of functions…

A: We will prove the given statement.

Q: 2) Use the Wronskian to show the three functions are linearly independent. f(x) = x, g(x) = cosx,…

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Q: 2. Consider the space L ([1,2)) of complex square integrable functions on the interval [1, 2).…

A: Consider the  equation

Q: If z = f(u) + g(u), where u = - and v = x y. Prove that y %3D x: Zx - y· Zy = y2 · Zyy – x2 . Zxx…

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

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

A: Concept:

Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: d x"J,(x)=-x"Jns(x) d, n+1 X.

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Q: Consider P2 with the inner product (p, q) = | | p(x)q(x) dx . Let U = span{1, 2x + 2} and g(x) = 3x2…

A: See the detailed solution below.

Q: If f(x, y) is a potential function for the gradient vector field F(x, y) (Зх — 2у) і — (2л — у) ј, —…

A: Since F is conservativei.e, ∇×F  , so there exists some potential function f so that ∇f =F .

Q: If z = f (u) + g(v), where u = = and v = x ·y. Prove that Q1: x· Zx – y· Zy = y² · Zyy – x² · Zxx…

A: We have z=fu+gv, and u=xy  and v=x.y We need to use the chain rule and prove x.zx-y.zy=y2.zyy-x2.zxx…

Q: 3. Let C[0, 1] be equipped with the inner product (f. g) = f(x)g(x)dx, where f = f(x) and g = g(x).…

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Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: If z = f(u) + g(u), where u = = and v = x · y. Prove that x · Zx – y · Zy = y² ·Zyy – x².Zxx using…

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Q: The domain of the function f (x, y, z) = In(3ry – 6) is %3D - O {(x, y) E R² : xy 2} o {(z, y, z) E…

A: Given function is: f(x,y,z) = ln(3xy-6) We need to find the domain  for this function.This can be…

Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: Q3: a) Let ƒ : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its

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Q: A function f(x,y) is homogeneous of degree 7 in x and y if and only if, __________.

A: Given problem is solved below.

Q: 4. Show that the map K: [1, o) → [1, ∞0) defined by 1 t dt K(x) = 3 e 2 is a contraction in the norm…

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Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: Suppose w=x/y+y/z, where x=e3t, y=2+sin(t)x=e3t, and z=2+cos(6t). A ) Use the chain rule to find…

A: w=xy+yz, where x=e3t, y=2+sint & z=2+cos6t Partially differentiating w=xy+yz with respect to x,…

Q: (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by…

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Q: Let AC C([-1,1]) be defined by A = {f€c'(I-1, 1), f(0) = 0, f'(x)| < 1, V x€ (-1,1)} Prove that A is…

A: The set A is defined as A=f∈C1-1,1, f0=0, f'x≤1, ∀x∈-1,1. To prove that A is relatively compact.…

Q: use the inner product = p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

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Q: Let A ∈ M nxn(F). Show that A is diagonalizable if and only if At isdiagonalizable

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Q: If z = f(u) + g), where u = = and v = x· y. Prove that y x: Zx - y Zy = y? Zyy – x2 - Zxx using the…

A: We have to prove this

Q: 3. Prove by using the Squeeze Theorem, or otherwise: 1 t2 dt is continuous at 0. T sin t (a) (b) 1+…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

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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: Let {yk } be the DFT of {x; }, where x = [0, –1, 0, 1] y3 = O i -1 -i 1

A: Here, x0=0, x1=−1, x2=0 and x3=1 We have to find y3 Now, y3 is given by…

Q: If the Wronskian of f and g is t cos t - sin t, and if u = f+ 2g, v = f - g, find the Wronskian of u…

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Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

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Q: 2. Let z = sin(xy) + x siny,x=u² + w² and y = uw. 2a. Compute Vz= when u = 0 and w = 1.

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Q: (1) Show that F(x, y) : (2x + y, x - y) is path independent and use the fundamental = theorem of…

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

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Q: Find the divergence of the field F. F = yj-xk (y2, x2 1/2 x2- y2 (y2+ x23/2 -x2 ( 2+ x33/2 x2…

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Q: The curl of f(x,y,z)=xyzj is None of these. хуk-yzi О ху-yzk ООО О

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Q: Find the Jacobian of following functions f(x,y, z) = 3xy – yz + 2y z2 - 5xyz3 g(x,y, z) = xyz -…

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Q: Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t),…

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Q: 4. Show that the function f : X → Y defined by f(X) = X U {a} vX €x is a bijection.

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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

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Q: 2. Let f(t) be a continuous function on the interval [0, 27]. Suppose the 1st order Fourier…

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