12) Prove that if a and b are constants and f and g are scalar fields, then V(af bg) aVf bVg

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Q: Prove that any vector field, ∇.( ∇ XE)=0.

A: To prove divergence of curl is  zero. Let us define a vector valued function: K=Kx(r), Ky(r), Kz(r)…

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Q: Prove that if A is invertible and diagonalizable, then A-1 is also diagonalizable.

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Q: If H is a Hilbert space and A € BL (H) is positive, prove that A+21 is invertible.

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Q: 1. Prove that Se A(V) is regular if and only if whenever v1, ..., vn e V are linearly independent,…

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Q: QUESTION 4 In a vector space V, prove that Ov=0 for all veV.

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Q: Find the degree and a basis of the extension field Q(v3, v7) Q(V3 + v7). over

A: This question is from field theory's topic extension of field. Solution is given below.

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Q: 4. Prove: If b is a non-zero vector, and x and x are both solutions to the matrix equation AX =6,…

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Q: Prove that if A is diagonalizable, so is AT.

A: Given: Matrix A is diagonalizable. To prove: AT is diagonalizable. Concept Used:1) If a matrix A is…

Q: Show that if A is both diagonalizable and invertible, so is A-1.

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Q: Prove that cond(A) = 1 if and only if A is a scalar multiple of a unitary or orthogonal matrix.

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Q: 1. Suppose U and V are n-dimensional vector spaces over the same field F. Prove that U and V are…

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Q: 4. Let ƒ,g be C² scalar fields on R³. Verify that V x (ƒVg) = Vƒ × Vg. V × (ƒVg+gVf). V· (SVg) =…

A: As give by the question There are given that the scalar field is f, g be C2 on ℝ3. And, In the above…

Q: 1. Prove that V is a vector space. V = pn , the set of polynomials with real coefficients and any…

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Q: 1. Suppose V is a vector space over C and D: V x V → C is a sesqui-linear form. Show that is…

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Q: -2t y= c1e -1 1

A: For the solution of the problem follow the next step.

Q: 7. If f is a scalar field and v = [v1, v2, v3] is a vector field, show that div(fv) = fdiv(v) +v •…

A: We will use simple definition of divergent of vector quantity and gradient of scalar quantity.

Q: Determine a scalar r such that Ax = rx, where 1 A =| 1 1 and x = 4 -4 5

A: Ax=r x ⇒ x is an eigenvector of Afor the eigenvalue r. So to find r, we compute all the eigenvalues…

Q: State whether the following statement is true or false in a vector space over a field E: " a,BEF and…

A: Given that , V is a vector space over a field F.

Q: 5. Prove that if F, G are vector fields on R3, then curl(F + G) = curl(F) + curl(G).

A: Let the vector fields F and G are: F=Fxi+Fyj+FykG=Gxi+Gyj+Gzk Then,…

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: Prove that if u is orthogonal to both v and w, then u is orthogonal to sv+tw for all scalars s and…

A: Given that u is orthogonal to both v and w.

Q: Suppose a is a non-zero scalar, v = (-a, 2a, a), and w = (a, -a, 2a), which of the following equals…

A: Given : v = -a, 2a, aw = a, -a, 2a

Q: Let A be an n × n matrix and let λ be a scalar. Show that det(A − λI) = 0 if and only if Ax = λx for…

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Q: 4. Find representation of the vector v = 2 – 2x+ 6x? with respect to following basis. {1+ 3x + 3x?;…

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Q: 1.21. Find the relationship which the cartesian components of A and B must satisfy if the vector…

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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 A and B benxn matrices such that AB = 1. Prove that |A| +0 and |B| + 0.

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Q: Prove that Vn(I) is a vector space

A: Prove that VnI is a vector space.

Q: Let u,v,w be linearly independent over a field F.  Show that{u+v,u−2v,u−v−2w}is also linearly…

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Q: Prove that, if A is similar to B, and B is diagonalizable, then A is diagonalizable.

A: Given that A is similar to B and B is diagonalizable.  To find A is diagonalizable.

Q: 7. Prove that M2x2(C) is a complex vector space under the usual addition and scalar multiplication…

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Q: Give a non-zero vector in the null space of A. 4 A = -4 -5 2 4-3 -55-3 -1 1-5 -3.

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Q: Show that if Aij is symmetric and Bij is skew-symmetric, then the scalar η given by η = AijBij is…

A: A tensor aij is said to be  symmetric if aij=aji. Similarly a tensor bij is said to be anti…

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Q: Let A = (A, A,y, A2) and B = (B,, By, B2) be two arbitrary vector fields, prove the following…

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Q: Prove that if A is invertible and BA = CA, then B = C.

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Q: 6. Suppose that V is a vector space over a field F and that v1, v2, ..., Vn E V. For each statement…

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Q: 7. If U¡¡ ‡ 0 are components of a tensor of the type (0, 2) and if the equation ƒU¡¡ + gU ji ³ = 0…

A: Given : Uij≠0    ,fUij+gUji=0 To Prove : f=g ,  Uij  is skew symmetric

Q: Show that if k is a scalar and A is n × n, then det(kA) = kn det(A).

A: Show that if k is a scalar and A is n × n, then det(kA) = kn det(A). For any scalar k, we need to…

Q: Determine which set of vectors form an orthonormal. a) b) c) d) None of these 11L

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Q: If p (x, y, z)s a scalar field, which of the following are valid operations? 1. V (V · Vy) 2. V x (V…

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Q: 7. If f is a scalar field and v = [v1, v2, v3] is a vector field, show that div(fv) = fdiv(v)+ v ·…

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Q: Sketch the following vector fields: 1.

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Q: Show that the vectors u = (1+i, 2i) and w= (1, 1+i) in C² are linearly dependent over the complex…

A: The vectors u and w are linearly dependent in K2 (with K = ℝ or ℂ), if there exists a,b∈K with a=b=0…

Q: Determine a scalar r such that Ax = rx, where

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