32. In a normed vector space (V, ||) show that B,(x) = x + B,(0) = {x + y: Ilyll

Q: Let A - ( ) a. Show that the vectors lz.A, A² are linearly dependent in Maz (R). b. Deduce that A"…

A: Statement (S1) proves that  I2 , A, A2  are linearly dependent.

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

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Q: What is the dimension of the vector space M,x3R)?

A: SOLUTION-

Q: If V is a vector space than show that cV=V

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Q: 8. Compute the flux of the vector field. F(x,y,z) = (2x -z,-2y, 1-2x) through S= {(x,y,z) € R³: z =…

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Q: In order for vectors and t- 3 in a vector space R3 to be linearly t+4 in depen den t what should t…

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Q: Suppose V1, ..., Vn are vector spaces then does L(V1 x...x Vn, U) have the same dim with L(V1, U)…

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Q: 3. Compute the scalar line integral 3 y – xz ds where C is the line seg- ment from (0, 1, 1) to…

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Q: H.W: Let V = {(x, y): x 20} with standard operation on R2. Is Va vector space?

A: A vector space is a set V that satisfies the following 8 conditions. Here two operations "+" and "."…

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: 6. Consider a normed vector space S = (R,|- |). Show that, for all r, y e S, |z - y| 2 | – l9|-

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Q: What is the dimension of the vector space S=abc detEM2x3(R): a=b=c=0}

A: Dimension of vector space is number of element in basis

Q: Let V = {(x, y): x 20) with standard operation on R2. Is V a vector space?

A: Given set is  V={(x,y):x=>0 }

Q: Consider the vector space Rn with inner product{x, y} = xTy. Show that for any n × n matrix A,(a)…

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Q: Let r = (2,-i, 1), and y = (-i, 3,1 + i) be vectors in C3. Compute (x, y), ||x||, |ly|l, and…

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Q: 5. Compute div F and curl F if F = (2x – cos y)i – x²e32j + (z² – 7x)k. 6. Determine if the vector…

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Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…

A: The vector field is,F¯=6x2+2xy, 2y+x2z, 4x2y3The region D is cut from the first octant by the…

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: (6) Suppose that T is a linear operator on a 2 dimensional vector space V and that T+ a I for any a…

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Q: 2. Let W be the union of the first and third quadrants in the xy- plane. That is, let W = xy 0 a. If…

A: According to our company gardening can answer only one question. If a specific questions are asked…

Q: Let V ((x,y): x 20) with standard operation on R2. Is V a vector space?

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Q: 4.) Verify the Divergence Theorem where F(x, y, z)=(x',xz,3z) given by x' + y +z² <4. and E is the…

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Q: Recall that we say that a random variable X is in the vector space L^2 if it has finite second…

A: Properties of a norm:In order to prove that ||X||2 = √E(X2) is a norm, it must satisfy 3…

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: Let V = {(x, y): x 20} with standard operation on R². Is V a vector %3D space?

A: Our objective is to conclude V is a vector space or not.

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

A: Given : V is any normed vector space and x is any non zero vector and α&gt;0. To find :  A real…

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: Show that F = (3, 1, 2) is conservative. Then prove more generally that any constant vector field F…

A: Given: F = (3, 1, 2) To show: F is conservative. Also to show any constant vector field F = (a, b,…

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: Let V = {(x, y): x 20} with standard operation on R2. Is V a vector space?

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Q: 11. Let (V.||-|) be a normed vector space. Suppose e : V - R has the property that e(r+y) =…

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Q: Let V = {(x,y) where y= 3x+1 x ER} with scalar multiplication defined by k(x,y)=(kx,y) is a vector…

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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: 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: 5. Show that there is no vector field G such that: V x G = 2xi + 3yzj – xz?k.

A: Given that ∇×G→=2xi^+3yzj^−xz2k^. We have to show that there is no vector field G→ such that…

Q: Theorem 2: Let X be a finite dimensional normed space and let r > 0. Then, the closed unit ball B[0;…

A: Theorems and definitions we are going to use: 1. Definition: (X,|| . ||) is said to be bounded if…

Q: 5. Is the vector field F(x, y, z) = (x(z² – y²), y(x² – z²), z(y² – x²)) incompressible? Justify…

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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: ƏF dF = (dr. V)F+ -dt. at

A: Given: A vector function F depends on both space coordinates x, y, z and time t.  To Find:  To show…

Q: Which one of the following vector fields is conservative? O F(x, Y, z) = (xy +1, a²y? – 2, xyz)…

A: 1) fx,y,z=xy+1,x2y2-2,xy2 Let fx,y,z=xy+1   yx,y,z=x2y2-2 hx,y,z=xyz The curl of Fx,y,z = ∇xFx,y,z…

Q: 6. Show that the system with vector field -x + sin(y) F(x, y) : -y+ cos(x), admits no periodic…

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Q: 14. If d is a metric on a vector space X‡ {0} which is obtained from a norm, and å is defined by…

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Q: 1) Verify that d/dx(fg) is a vector space over R, or explain why it is not.

A: The objective is to find whether the ddxf.g is a vector space over ℝ. Since it is known that…

Q: If a vector field A(x,y,z)=xz3ex - 2x2yzey+2yz4ez , calculate the divergence ∇·A at point…

A: A vector field is given as  and

Q: : Verify the divergence theorem for A=4xi +(2Y+z Vector field ) | +(52-2x)k in the regim beunded by…

A: According to the given information, it is required to verify the divergence theorem for a vector…

Q: 2. Verify that the usual length of Vector in the has the plane properties in 3 dinensional Space…

A: Given: usual length of a vector in the plane or in 3 dimensional. To prove that this vector has the…

Q: 3 Let W = span Given y = 1 decompose this vector into y in W and y, in w. 3 13-

A: See the detailed solution below.