- Let T : R² → R² be defined by T -E} Let u = and C = Vi = V2 = Find T(u), the image of u under T. Find [T(u)]c, the coordinatization of T(u) with respect to the basis C. Ex: 5 T(u) : [T(u)]c =

Q: Determine the divergence of vector field E = 3x²x + 2zŷ + x²z2 at (2, –4, 0).

A:

Q: Let C := {1, cos t, cos 2t, . , cos 6t} C V where V is the vector space of all real-valued functions…

A: Given: C:=1, cost, cos2t,…,cos6t⊂V. V=f | f:0,2π→ℝ is the Vector space. J=spanC To prove: C is a…

Q: Find a basis for the plane 4 + 3y + 5z = 0 in R. %3D

A:

Q: 1) Find the work done by the vector field F(x, y, z) = (x², 4xy³, xy²) on a particle that %3D…

A: Given force is F→x,y,z=x2,4xy3,xy2 Let r→=rt→ be the boundary of the rectangle with vertices  0,0,0,…

Q: Show that the vectors (3+ /2,1+V2) and (7, 1 +2\/2) in R² are linear!ly dependent over R but…

A:

Q: V.(V×F)= 0

A: Now given, ∇.∇×F=0 To prove the above.

Q: What is the divergence of the vector field f = x²i+ xyzj – zk at the point (1, 2, 3). Select one: а.…

A: f=x2i+xyz j-zK at (1,2,3)  div f=i ∂f∂x+j.∂f∂y+K.∂f∂z

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: Let P, have the inner product (p,q) = | p(x)q(x)dx. Apply Gram-Schmidt process to transform the…

A:

Q: Find a basis of R that contains the vectors X = (1,1,1,1) and Y = (1,1,0,1).

A: Given Vectors X =(1,1,1,1) and Y = (1,1,0,1) Now determine the orthogonal basis for R4

Q: (a) Show that the vector field F (x, y, z) = (x² + y², x, z³) is not conservative.

A:

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

A:

Q: Find the divergence of the vector field F→(x,y,z)=xyzi→−yzj→+xzk→at the point (8,3,2) .   Enter…

A: The given vector field is : Fx,y,z=xyzi-yzj+xzk To find the divergence at 8,3,2

Q: Compute the flux of the vector field (x, - xy*), out of the rectangle with vertices (0,0), (5,0),…

A:

Q: ,Let Pn be the vector space of all polynomials of degree n or less in the variable r. Let D : P3 →…

A:

Q: Find the matrix for T relative to the basis B, and use Theorem 8.5.2 to compute the matrix for T…

A:

Q: The Gram-Schmidt process transforms the basis u,=(1, 3), u,=(1, –7) of R² into the orthogonal basis…

A:

Q: Consider the space of square integrable functions L² ([–1, 1]). Let B = {Pn|n = 0, 1, 2, ...} the…

A:

Q: 3. Let F be the vector field (ze²z + yz², xz², xe*z + 2xyz)

A:

Q: Find vectors x and y in R2 that are orthonormal with respect to the inner product (u, v) = 3u₁v₁ +…

A: The given problem is related with inner product. We have to find the vectors x and y in ℝ2 that are…

Q: Compute the flux of the vector field (x, (æ°, – ay*), out of the rectangle with vertices (0,0),…

A: To compute the flux of the vector field x3,-xy4, out of the rectanglewith vertices 0,0,2,0,2,4…

Q: Consider inner-product =| f(x) g(x) dx in vector space C[- n, 7], the Fourier expansion of f(x) = x…

A:

Q: 15. Use the Gram-Schmidt process to transform the basis {r1, x2, a3, T4} into an orthogonal basis…

A:

Q: Find the curl of the vector field F(x,y,z) = x²zi – 2xzj + yzk and the point (2, -1, 3).

A: Let the given vector field is :

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: Sketch the vector field on given byF(x, y, z )= z K

A:

Q: Show that the vector field F(x, y, z) = (2y cos(-3x), –3x sin(2y), 0) is not a gradient vector field…

A:

Q: Define T: R-R by Tix) =Ax, where Ais the matrix defined below. Find a basis B for R? with the…

A:

Q: Let uj = (1,1,2), uz = (1,3, –2), uz = (4, –2, –1) in R’. Show that uj , uz, Uz are orthogonal, and…

A: The answer is given in this below photo

Q: Let {u1, u2, u2} be an orthonormal basis for an inner product space V. Suppose v = au1 + buz + cu3…

A: Topic:- vector algebra

Q: The follow ing vectors {, x,, x3}are linearly independent. Construct an orthogonal basis for w…

A:

Q: The Hermite polynomials form an orthogonal (not orthonormal) basis with this inner product: (Hm[Hn)…

A: Given:                                   fx=x3…

Q: 6. Vector fields V and W are defined by V = (2x – 3y + z, –3x – y + 4z, 4y + z) W = (2x – 4y – 5z,…

A:

Q: The Gram-Schmidt process transformsthe basis u1=(-3,1), u2=(7, 1) of R´ into the orthogonal basis…

A: Orthogonal set problem....

Q: Compute the flux of the vector field (x3, – xy³), - out of the rectangle with vertices (0,0), (3,0),…

A: Given: F=x3 ,-xy5with rectangle with vertices (0,0) , (3,0) , (3 ,2) , (0,2) To find flux of vector…

Q: Let B = {1, x, x², x³, xª}, which is a basis for the vector space of polyno- mials W = P4(R). Let V…

A: Let the best approximation of the function f(x)=sinx+cosx is g(x), where  g(x)=∑i=15f(x),eiei…

Q: Let F(x,y,z)=(2xz^2,−2xyz,9xy^3z) be a vector field and f(x,y,z)=x^3y^2z (∇f=(  ,  , )). (∇×F=(  ,…

A: Given:F(x,y,z)=2xz2,-2xyz,9xy3zf(x,y,z)=x3y2z

Q: Show that V x VG = 0 for all scalar field G(x,y,z). %3D

A:

Q: 70. Verify that T(x, y) = (2x +iy, ix + 2y) is a normal operator on C². Find an orthonormal basis B…

A:

Q: 2. Let the inner product on vector space V be (u,v)= uvdz. Use the Gram- Schmidt process to…

A:

Q: Consider inner-product = | f(x)g(x) dx in vector space C[- n, T], the Fourier expansion of f(x) = x…

A:

Q: Let {u₁, U2, u2} be an orthonormal basis for an inner product space V. Suppose v = au₁ + bu₂ + cuz…

A: We have v=au1+bu2 +cu3 Now, ∥v∥=a2+b2+c2 Given ∥v∥=80 So,  80=a2+b2+c2⇒802=a2+b2+c2⇒80=a2+b2+c2…

Q: Theorem 7. Let T be a linear transformnation from space U (F) into a vector space V (F). Then T is…

A:

Q: The Gram-Schmidt process transforms the basis u=(1, 2), u2 =(6, 2) of R² into the orthogonal basis…

A:

Q: 7. Consider the basis B-{v,,Vz} for IR? where Vi =(I,1) and Vz =(-1,1). Let T:IRIR? oe a linear…

A: NOTE: Refresh your page if you can't see any equations. .

Q: Show that Green's Theorem holds for the vector field F(x, y)=(x +y, 3x) at a square of side 2.

A:

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

A: Given: The vector field F=Pi+Qj+Rk is conservative P, Q, R have continuous first order partial…

Q: 3. Show that the vector field F = -C with C> 0, is conservative and find its potential.

A:

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

A: