5. Let V be an inner product space, and T E L(V). Prove that null(T) = (Im(T*)). %3D

Q: 7. In A ABC,X is the centroid. A A. If CW = 15, find CX and XW. W B B. If BX = 8, find BY and XY. C.…

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Q: 6. Evaluate (7xi - zk) · ndA over the sphere S : x² + y² + z² = 4 (u) by the Divergence Theorem, (b)…

A: The divergence theorem states that  ∬SF·ndA=∭V∇·FdV Where, F is the vector field, V is the given…

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A: (Solving the only one question as per our guidelines) Question: Let x∈ℂm and y∈ℂm Show that x,y=y*Ax…

Q: 10. Let X be an inner product space and T: X→→→→→X an isometric linear operator. If dim X<∞, show…

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Q: The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of…

A: We will use the basic knowledge of multivariate functions to answer this question. Note that in your…

Q: nclude spaces in between 5. d te In (In y)) at y dy

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Q: 1. Let d, be a metries on Xo and let the function a: X x XR defined be dr. y) dis. y)= prove that…

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Q: Let F(x, y, z) = yi – xj+ za*y² k. Complete each of the following. (a) Compute the curl and…

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Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

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Q: 5. Evaluate Fale JJJ (²+ + ² dv, where a, b, c > 0, and & is the region given by E 22 + y² 2² 6² +2…

A: Given triple integral is ∫∫∫Ex2a2+y2b2+z2c22dVa,b,c&gt;0x2a2+y2b2+z2c2≤1 in the first octant. Now,…

Q: and y = 2u, evaluate the integral xy³dA where c) Using the transformation x =-; 6u region R is…

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Q: 3. Show that every y E lị defines a continuous linear functional on l through the inner product,…

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Q: Verify Green's Theorem by evaluating both integrals | y2 dx + x2 dy dA ду for the given path. C:…

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Q: Use the definition of the dot product a b = |a||b| cos 0 to prove the Cauchy-Schwarz Inequality: |a…

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Q: Let T:V U be a homomprphism then ker T={0}if T is one-one.

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Q: 6. Let D be the region in the ryz-space defined by the inequalities 1 <z< 2, 0 < ry < 2, and 0 < z <…

A: Given integral, ∫∫D∫x2y+3xyzdxdydz where  1≤x≤2, 0≤xy≤2, 0≤z≤1 and transformation u=x, v=xy, w=3z…

Q: Suppose that Di is the open disk with center 0 and radius 1. Let f(z) be analytic on Di \ {0} and…

A: Given- Suppose that D1 is the open disk with center 0 and radius 1. Let fz be analytic on D1\0 and…

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: (6) I. adv, where E is bounded by the planes z=0, y = 0, =0, y+= =1 and +=1, as shown in the figure…

A: Given that,

Q: Let I = , zdV ,where D is bounded by the planes x = 0, z = 0, x+ y = 2 and the cylinder x2 + z2 = 1…

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

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: Consider the region RR in the xyxy-plane that is described by the intersection of the four lines:…

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Q: In the space C[0, 1] of continuous functions on the interval 0 <r < 1, with the inner product (f(x),…

A: We shall use the properties of inner product and the basic formula to find the projection of a…

Q: 6. Consider S = [a, b] with p(x, y) = |x – y|. Verify that T : S – S with T(x) = - and a > 1 is a…

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Q: let X-j m,n; is Z Ž asigma algebra over X ?

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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: Evaluate ∫∫∫E √x2 + z2dV, where E is the region bounded by the paraboloid y = x2 + z2 and the plane…

A: Given, y=x2+z2 ⇒z2=y-x2z=±y-x2 Therefore the limit of z is -y-x2≤z≤y-x2 On drawing the graph in the…

Q: Let M be an elliptic paraboloid a = y? + z². Compute the Gauss map and the shape operator. Find k…

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Q: 2. In A ABC, X is the centroid. A A If CW = 15, find cx and XW. W, B. If BX = 8, find BY and XY.

A: we know that  centroid is located 23 of the distance from the vertex along the segment that connects…

Q: Use Greens Theorem to evaluate: §. (2x + ev´)dy - (4y2 + e*")dx where C is the boundary of the…

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Q: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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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: For z ∈C, define Tz: C →C by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Firstly, find out Tz*.

Q: Let f(x,y,z) = zIn(x³ycos(z)), find Z 3x² O In(x3ycos(z)) - ztan(z) 0 2 Y о 3z X af ila

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Q: 4. Show that ||y|| = maxxela.b]ly'(x)| defines a norm for the linear space Y - %3D {y e C' [a, b]: (…

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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: Find the kernel of linear map L:R R given by L(x,y,z)%3D(0,z,y)?

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: Verify Green's Theorem by evaluating both integrals aN aM y2 dx + x2 dA ду ax for the given path. C:…

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Q: Apply Gauss Divergence theorem for F = x²i + y²j + z²k_over the cuboid S bounded by 0 < x< 1, 0< y<…

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Q: Let R be the region in the first quadrant bounded by x = = 1, (x − 2)² + y² = 1, and x² + y² = 4.…

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Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

A: In this question, we use the given sketch of the surface and find the parametrization for each…

Q: b. Draw the image of the W-plane into which the region bounded by the straight lines in the Z-plane…

A: The solution is given below

Q: Use Green's Theorem to evaluate S. axy dy + cxy dy, where C is a counterclockwise oriented…

A: Given : a=2, c=8 We have to use Green's Theorem to evaluate            ∫C axy dy+cxy dy where C is a…

Q: [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…

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Q: Evaluate / (7xi – zk) · ndA over the sphere S : x² + y² + z² = | S a) by divergence theorem, (b)…

A: a) By divergence theorem, ∬SA. ndA=∫∫∫V∇.A dV Now,  ∇.A=∂7x∂x+∂0∂y+∂-z∂z=7-1=6 By changing to…

Q: b) Let f(x) = 1 and g(x) = x be functions in C[0, 1]. Use the inner product defined in Example 5, to…

A: Note: Since there is no proper instruction is given and the first part is incomplete, we have solved…