Question Let X, Y be the two vectos spacces over the Same field F andy T: X→Y is a funchion such that T(ax+by)= aT (X)+bT(y): Then %3D li)RIT)=T(X) = T(x): xEX}is subspace of Y %3D ü)N(T)= ker(T) ={x€X:T(x)=0}is a subspace of X:

Question Let X, Y be the two vectos spacces over the Same field F andy T: X→Y is a funchion such that T(ax+by)= aT (X)+bT(y): Then %3D li)RIT)=T(X) = T(x): xEX}is subspace of Y %3D ü)N(T)= ker(T) ={x€X:T(x)=0}is a subspace of X:

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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*Q6: [Exercise 33] For each of the following vector fields, find the divergence: (i) v=xi+yj+zk, (ii) v = x² i + y² j + z² k, (iii) v=yzi+xzj+xy k, (iv) v = sin(x) i + cos(y) j - z(cos(x) = sin(y)) k. Which of these vector fields (if any) are solenoidal?
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Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•₁•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x, x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (x, ay) = a(x, y) for all x, y V (iv) (x, y+z) = (x, y) + (x, z) for all (v) (x, y) = (y,x) for all x, y V. (a) (.,.): R¹ × R¹ → R is defined as xεν. (x, y) = x¹y, and for all x,y,z € V. Example: Verify that each of the following mappings define a real inner product. a EF. for all x, y ER".
7. Let F = F i + F25 + F3 k be a smooth vector field and let f(x, y, 2) be a smooth scalar function. Match the follwing af af + (a) dz ƏF3 (b) dz (c) dx dy dz F1 F2 (d) Vƒ (e) V.F (f) ▼ × F with one of (i) divF (ii) curlF (iii) grad f
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