1. Consider the real vector spaces C2 and R2×2, and let f: C2 → R²×² defined by f(x, y) = (4 'а — d a +c b+d. с — b ),v(x, y) = (a + bi, c + di) E C². (a) Prove that f E L(C², R²×²).

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: I can't find C, D, and E

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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: 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: Given a vector field F = 5(x² + y²)i+ 2xyj - 3zk, evaluate SSS, V· FdV for which the solid D is…

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Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

Q: Consider the autonomous system of non-linear equations * = u(x, y), where u is a vector field with…

A: In this question, the classification of the equilibrium point is done.

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: 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: 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: ind the flux of the vector field F = yzi + 7yj + yk over the sphere x2 + y2 + z2 = 4.

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Q: What is the dimension of the vector space ab c S= { EM2x (R): a =b=c} de f

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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: 2. Show that any vector field of the form F(r, y, z) = f(y, z)i + g(x, z)j+ h(x, y)k is…

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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: Show that the vector field F (x, y, z) = xz i + xyz j y 2 k is not conservative.

A: Consider the following vector: Fx,y,z=xzi^+xyzj^+y2k^ Find the Curl of F→:…

Q: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (x, , y.) +…

A: We have to show that the set V with the addition operation x1, y1+x2, y2=x13+x233, y13+y233 and…

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

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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: Show that the field R(i) as a vector space over R has a basis { 1 , i } and is isomorphic to  R_2,…

A: The field ℝ (i) is defined as ℝ (i) = {a + bi| a, b ϵ ℝ}.That implies, ℝ (i) = {a × 1+ b × i | a, b…

Q: Suppose that W is a finite-dimensional vector space over F and that T:V→W is linear. Prove that…

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Q: Show that V={(x,2020x): x in R} is a vector space

A: According to the given information, it is required to show that the given set is a vector space.

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>0. To find :  A real…

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: 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: 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: if it was M, N,L Subspaces of the vector space v on the field F Prove that (Mn N) + (M n L) c M n (N…

A: Given that M, N, L are subspace of a vector space V over the field F. We have to prove that M∩N +…

Q: -dt. dF = (dr · V)F+ at

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Q: Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous…

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Q: Let V and W be vector spaces and T:V→W be linear. Let {y1,…,yk} be a linearly independent subset of…

A: It is given that Let V and W be vector spaces and T: V→W be linear. Let {y1… yk} be a linearly…

Q: Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x² +…

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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: (10) Given that the vector field F= 2z i- 3 x j + 4 y k Evaluate f F. dr where S is the portion of…

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Q: Consider vector space V5 over field F with dimension 5. Consider linear mapping A E L(V5).…

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Q: Let H be the set of all points in the second quadrant in the plane V=R^2. That is, H={(x,y) | x ≤ 0,…

A: Here in the question H be the set of all points in the second quadrant in the plane V=R2. That is,…

Q: c) Prove that a subset W of R’ defined by W ={(x, y, z) e R² / 2x= y = 2z} is a vector space of R'.

A: Given, W = { (x,y,z) in R³ | 2x = y = 2z } is a subset of R³ Let, u = (x,y,z) & v = (a,b,c)…

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: Let F(x, y) = (2x + y³, 2y – x4). Compute the flux oF. F.n ds of F across the circle of radius 5…

A: Given function can be written as, Region bounded by the circle of equation,

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: 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: 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: 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: 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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