let V be the set of all pairs (x,y) of real numbers and let F be the field of real numbers. define (x,y)+(x1,y1)=(3y+3y1,-x-x1) c(x,y)=3cy,-cx Verify that V with these operations is not a vector space over the field of real numbers.

Q: Show that the vector field F(x, y, z) = xz i + xyz j - y 2 k can’t be written as the curl of another…

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Q: Let V be the vector space of functions from R into R. Show that f,g,h EV are independent where: f(x)…

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Q: Let V be a vector space over a field F and let w, x, y, z ∈ V. Show that w + (x + (y + z)) = ((w +…

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Q: The vector field F is shown in the xy plane. 3- 2- 2 3 4 Is div F positive, negative, or zero?…

A: Divergence is a operator which operate on vector field F to gives a scalar quantity . And divF is…

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Q: Let V be a General vector space over F. Use the definition of General Vector Space, prove that V r E…

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Q: Show that if a vector field has the form F = ⟨ƒ(y, z), g(x, z), h(x, y)⟩, thendiv F = 0.

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Q: Let R be the field of real numbers. Then {(x,x,x): x ER}is a subspace of R (R) ?

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Q: F  is the three-dimensional vector field defined by F(x,y,z) = ( z, x, y ).  In other terms,…

A: Consider the given information.

Q: Let C be the straight line segment in the xy-plane from the point (1, 2) to the point (5, 3). Let F…

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Q: Compute and sketch the vector assigned to the points P = (0, 1, 1) and Q = (2, 1, 0) by the vector…

A: Solution:Given       P = (0, 1, 1) and Q = (2, 1, 0)F=<xy,z2,x>To sketch the graph of F

Q: Let V be the vector space of set of all real polynomials over the field of real numbers ℝ. Let W be…

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Q: Let F(x,y,z)=(−6xz2,6xyz,6xy3z)F(x,y,z)=(−6xz2,6xyz,6xy3z) be a vector field and…

A: Given, Fx, y, z=-6xz2, 6xyz, 6xy3zfx, y, z=x3y2z

Q: Let V be the subspace of the vector space of all real-valued functions that is spanned by the setS…

A: If V is the subspace of the vector space of all real-valued functions that is spanned by the set S…

Q: Let V and V' be n and m dimensional vector spaces over a field FE T

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Q: let (x,y,z) be Scalar Field and F a vector field.. $ Derive expression for D. (OF) and * (OF) in…

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Q: Let v = [x2-y?, x2 +y², z] be a vector field. 1- Compute Curl(v).

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Q: Let V be the set of ordered pairs (x, y) of real numbers and let F be the field of real numbers.…

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Q: Sketch and describe the vector field F (x, y) = (-y,2x)

A: The given vector field is F→x,y=-y,2x A vector field F→ in ℝ2 or ℝ3 is an assignment of a…

Q: Use the definition of the divergence to show that div(c, F, + c2F;) = c, div(F,) + cz div(F,) for…

A: F1, F2 : D ➡️ R3 are differentiable vector fields and for constant c1 and c2 we have to prove that…

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: Determine whether the vector field given by F is conservative. F(x, y) = x2yi + xyj

A: Given vector field: F(x, y)= x2yi + xyj   Concept used: The vector field F = Pi+Qj is conservative.…

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: Let V be a finite – dimensional vector space over the field Fand i 1. 11 c 1 1 :

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Q: Let u, v be vectors in the space V over the field F and c a scalar. Prove that Span(u, v) = Span(u,…

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Q: Let v = [x2-2y², x²+2y², z] be a vector field. 1-Compute Curl(v). 2- Find the condition on x, y, and…

A: Curl of a vector.

Q: Suppose that f(x, y, z) = xy²z³ and F(x, y, z) = yzi+ zæj+ xyk are a scalar function and a vector…

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Q: The vector-valued function F (x, z) = (2 − x)i - Tx² kis an example of a vector field in the xz-

A: Given vector valued function F(x, z)=(2-x)i-πx2k

Q: Let V = {(a1,a2, ... ,an): ai ϵ C for i = 1,2, ... ,n}; so V is a vector space. Is V a vector space…

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Q: Compute and sketch the vector assigned to the points P = (1, 2)and Q = (−1, −1) by the vector field…

A: A vector field for two dimensional space is defined as a function that gives a vector for each point…

Q: Consider the vector space V = P over the field of real numbers F = R. Is the vector 4 – x – x 2 in…

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Q: Check whether or not the set V is vector space over R2 with respect to the indicated operations.…

A: To prove a vector space following axioms should be satisfied  for a set V with operations vector…

Q: 1) Let V be the set of all ordered pairs (x, y) of real numbers and F be the field of real numbers.…

A: Let V be the set of all ordered pairs (x,y) of real real numbers and F be the field of real…

Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the inner-product of…

A: We have to find the inner product of the following -  p = 2 + 1.7x q = 4.8 + 3.2x

Q: Let V be the vector space of polynomials of degree <3 over R. Determine whether u, v,w E V are…

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Q: Show that if a vector field has the form F = ⟨ ƒ(x), g(y), h(z)⟩, then∇ x F = 0.

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Q: Show that V² (v-x) = 2V.v+x-V²v, where v is a vector field and x is the position vector of a point…

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Q: Let v = [x2-2y, x² +2y², z] be a vector field. 1- Compute Curl(v).

A: Since you have asked multiple questions in single request.so we will answering only first question.

Q: (b) Verify Green's Theorem for a vector field F(x,y) = xy²i – yx²j on a disc with center (0,0) and…

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Q: 4. Let V be the set of all pairs (r, y) of real numbers, and let F be the field of real numbers.…

A: 4. Given that V be the set of all pairs (x,y) of reals numbersi.e V={(x,y):x,y∈ℝ}F=ℝDefine…

Q: The vector field F is shown in the xy plane and looks the same in all other horizontal planes. 3- Y…

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Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the magnitude of…

A: Given that the vector space is set of all polynomials of degree less than or equal to 1. For…

Q: Let v = [x²-2y², x²+2y2, z] be a vector field. 1-Compute Curl(v). 2-Find the condition on X, y, and…

A: If curl zero then irrotational

Q: Let f and g be real-valued continuous functions in the vector space C[a, b]. Show that (f. g) =…

A: Given that f and g be real-valued continuous function in the vector space Ca,b. We have to show that…

Q: Let V be a vector space with a scalar product. Show that = 0 for all v E V.

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Q: Give an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that…

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Q: There is a vector field F (x, y, z) such that curlF (x, y, z) = (divF (x, y, z) , (divF (x, y, z))²,…

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Q: x=y=z

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Q: Let C be the straight line segment in the xy- plane from the point (1, 2) to the point (5, 3). Let F…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.