Use the inner product (f, 9) = | $(2)g(x) dæ f(r)g(r) dr in the vector space C°[0, 1] of continuous functions on the domain [0, 1] to find (f, 9), ||f|| ||9||, and the angle afg between f(r) and g(z) for f(z) = 10z? – 3 and g(z) = -3z +1. (f, 9) ||f|| = |g|| = afg

Q: a) Find the divergence and the curl of the vector field F × G, where F(x, y, z) = 2x²yi – y³z²j –…

A: Given information: Fx,y,z=2x2yi⏜-y3z2j⏜-2z2k⏜ Gx,y,z=3xyi⏜-5yzj⏜+z2k⏜ The vector field F×G can be…

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B'A) Given the…

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Q: The given vector function is F(x, y, z) = (2x – y + In(z² + 1))î + (x – 6e²y + 4 cos 4y)ĵ + | 2xz -…

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Q: Suppose that the set A ={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n…

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Q: Consider inner-product = | f(x) g(x) dx defined for vector space C[-1, 1], then for the function…

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Q: If x, y and z are vectors in an inner product space such that (x, y) = (x, z), then y = 2.

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: Let W be the union of the first and third quadrants in the xy-plane. That is, let х :xy a. If u is…

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Q: Show that the vectors (3+ /2,1+V2) and (7, 1 +2\/2) in R² are linear!ly dependent over R but…

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Q: 3. Prove that if r, y E RP, then ||r + yl|2 = ||x||2 + |ly||² holds if and only if r y = 0. 4. Prove…

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Q: Consider the vector space V = C([0, 1]) of complex-valued functions that are continuous on [0, 1].…

A: To establish the Cauchy- Schwarz inequality for the functions in C[0,1]= the space of complex valued…

Q: If (a, b) is a local maximum of f, then D, f(a, b) = 0 for any unit vector u e R?. True False

A: Question: if (a,b) is a local maximum of f, then Du→f(a,b)=0 for any unit vector u→∈R2.

Q: Let ū = (u1, u2) be a unit vector in R2 and let f: R2 → R be defined by if (x, y) # (0,0), f(x, y) =…

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

Q: Find the divergence of the vector field. F(x, y, z)= 6x¹0i-10xy ¹0j O divF(x, y, z) = 60x⁹-100xy⁹ O…

A: To find the divergence of the given vector field.

Q: 7. Let C[-7,7] be the vector space of continuous function over [-n, 1] with an inner product 1 (5,9)…

A: C-π,π is a vector space of continuous function, with inner product <f,g>=1π∫-ππfxgxdx show…

Q: Let f(x) = sin x and g(x) = sin x + cos x  in the vector space C, [0, 2π] with the inner product…

A: The given functions are, In the vector space,

Q: Find the divergence of the vector field. F(x, y, z)= 7x¹i, 7xy7j O divF(x, y, z) = 7x³¡— Ţxy³j…

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Q: Let ū = (u1, u2) be a unit vector in R? and let f: R² → R be defined by if (x, y) # (0, 0), f (x, y)…

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Q: 17. If a vector function v(t) = (t, ln t, t-²) is given for a scalar function Ø = x Inz + √y-cos(x +…

A: We have to determine the line integral.

Q: Let f(x) = sin x and  g(x) = sin x + cos x  in the vector space C, [0, 2π] with the inner product…

A: Given that, f(x) = sin x and  g(x) = sin x + cos x Consider an inner product in them for f , g C ∈…

Q: Find the boundary of the sequence xn in the normalized vector space C [a, b], if it exists. 4 cos 3…

A: The given sequence is: xnt=4cosn3t5+cosnt a=-5π, b=-4π

Q: Consider the vector space C[-a, 7] with inner product / f(r)g(x)dr. (a) Show that the vectors f(x) =…

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Q: Use the inner product (f, 9) = | f(x)g(x) dr %3D in the vector space CU[0, 1] of continuous…

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Q: Use the inner product (f,9) = f(z)g(x) dz in the vector space C"0, 1] of continuous functions on the…

A: It is given that fx=-5x2-3 and gx=9x-7. We have to evaluate f, g, f, g and αf, g where αf, g is the…

Q: (d) Consider the vector space C0, 1] of all continuously differentiable functions defined on closed…

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Q: Consider inner-product = ] f(x)g(x) dx defined for vector space C[-1, 1] , then for the -1 function…

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Q: Find the boundary of the sequence xn in the normalized vector space C [a, b], if it exists. 4 cos/3…

A: Solution: Here the sequence is xnt=4cosn3t5+cosnt and a=-5π,  b=-4π

Q: O Show that whether the functions sint and cost in the vector space C[-n, 7]of continuous functions…

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Q: A vector field with component form F(x, y, z) = (P(x, y, z), Q(x, y, 2), R(x, y, z)) on R³ is…

A: Given that Fx,y,z=excosy,e-xsiny,2 The objective is to show that F is conservative. So to show that…

Q: Find the divergence of the vector field F. F(x, y, x) = i+ zj - 8y²zk %3D O2+ 7- 8y O zi+ 7zj - 8y²k…

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Q: Find the potential function for the vector field given as 8. F(x, y, z) = 4 O a. $(x, y, z) = x³y³z²…

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Q: . What is the basis and dimension of the vector space: SPAN {1, cos t, cos 2t}

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Q: In the vector space of all real valued functions, find O (1+cos 2t, cost sint, cos² t) O (sin 2t, 1+…

A: Given spanning set is sin 2t, 1 + cos 2t, cos t sin t, cos2 t

Q: -dt. dF = (dr · V)F+ at

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Q: Show that whether the functions sint and cost in the vector space C[-n, 1]of continuous functions on…

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Q: a) T: M, → R, where M, is the vector space ofn by n matrices over a field, and T (A) = det(A). b)…

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Q: Let F(x, y) = (4 cos(y), 2 sin(y)). Compute the flux $ F - n ds of F across the boundary of the…

A: By using Green Theorem, ∮F.n ds…

Q: What is the basis and dimension of the vector space: SPAN {1, cos t, cos 2t}

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Q: ) For the vector space C[0,1], define the inner product (f,g) = , f(x)g(x)dx. Let f = x + 1 and g =…

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Q: Let B = {1, x, x², x³, xª}, which is a basis for the vector space of polyno- mials W = P4(R). Let V…

A: Let the best approximation of the function f(x)=sinx+cosx is g(x), where  g(x)=∑i=15f(x),eiei…

Q: 5. If a vector function F depends on both space coordinates (x, y, z) and time t, show that dF = (dr…

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Q: Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that…

A: Given: The linear space C0,1 equipped with the norm, f=maxt∈0,1ft   To prove: There is no inner…

Q: Consider the curve C from (-2, 0, 1) to (6, 5, 2) and the conservative vector field F(x, y, z) =…

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Q: a) T: M, → R, where M,n is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

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Q: Let F(x, y) = (xy + 3x, xy-3y). Compute the flux § F · n ds of F across a simple closed curve that…

A: Given, Fx,y=xy2+3x,x2y-3y Compute the flux ∮F·nds of F.

Q: The vector field F(r, y) = (2xy +y) dx + (æ + 2xy) dy is conservative. %3D The function f(x, y) = x²…

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Q: Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary…

A: Given: The set A=x1,x2,...,xn, xi∈ℝn spans the entire space ℝn, i.e. for any arbitrary n-vector…

Q: Use the inner product 1 (f, 8) = f(x)g(x) dx in the vector space CO[0, 1] of continuous functions on…

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Q: The vector field A=xa,+y² a, +=' a, is given. Verify the divergence theorem for this vector over a…

A: Use the Divergence Theorem we  find the outward flux of the vector field F = x2i+y2j+z2k across the…

Q: Show that the vector field F(x,y,z)=yz^2i+xz^2j+2xyzk is conservative, find a potential function f,…

A: The given vector function is F(x,y,z)=yz2i+xz2j+2xyzk The given vector field F is conservative if…