(a)(3) Let V be a vector space over some field FLet W be a subspace of a vector space V. Suppose x, y € V are not in W.at if x = span (W, y) then y ≤ span(W,x)(b)Give an example of a vector space V, a subspace W ≤ V, and vectors x, y € Vsuch that x e span(W,y) but y & span(W, x)

Given: V be a vector space over field F To find: a) If x,y∈V W be a subspace x,y∈V is not in W…

Answered: (a) (3) Let V be a vector space over…

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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1. (Section 17.1) Match the vector fields with their sketches below by placing the letter of the function in the corresponding blank: (I) F(x, y) = (II) F(x, y) = (III) F(x, y) = (IV) F(x, y) = Vector Field: (B) Vector Field: (A) (C) Vector Field (D) 0 Vector Field:
Let V = R³, the vector space over R of dimension 3. (a) Give an example of a subspace of V. (b) Write down two different bases of V. (c) Show that the following is a scalar product (-, ·) on V : 3 a1 (u, 2) := a,b; v = | b2 b3, where U = a2 i=1 a3 (d) Give an example of a vector in V with length 5. (e) Use the Gram-Schmidt Process (or otherwise) to find an orthogonal basis for V containing 1 the vector wı
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Let V will be the vector space over R,L:V→V - linear mapping and v, w EV. Find L(v) and(w) knowing that a) L(2v +w)=v+wiL(v-w)=3v+2w.
Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u, w) where u E U and wE W. Show that V is a vector space over K with addition in V and scalar multiplication on V defined by (4, w) + (u', w') = (u + d , w + w') and k(u,w) = (ku, kw) (This space V is called the external direct produd of U and W.)
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1. Consider two vector fields: F1(x, y) = -yi+ xj and F2(7) = F. (a) Evaluate F and F, at the given points. (x, y) (1,0) (0, 1) (-1,0) (0, –1) F(r, y) F(1, y) (т, у) (1,1) (-1,1) (-1,–1) (1, –1) F(1, y) F(x, y) On the grids shown below, sketch above vectors of vector fields F (b) and F. F;(x, y) = -yi + xj F2(F) = F.
11. Let (V, (, )) be a complex inner product space. For vectors v, w, set (v, w) R = ¹/[(v, w)+(w, v)]. Consider V to be a real vector space. Is (V, (, )R) an inner product space? Support your answer with a proof.
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What are V-F and VX F for the vector field F(x,y,z) = 3x²zi-ye*j - y²zk? OV.F = x²exy 2 VXF = x²i-3e²j+yk OF=3x²-xe* + 2yz VXF = 2i-3j-k V-F = 3x²-x-z VXF= -2yi+ 3xj + zk V-F = 6xz-e* - y² VXF = -2yzi + 3x²j-ye*k
Which of the listed pairs has the set U as a vector subspace of the vector space V?

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(a) (3) Let V be a vector space over some field F Let W be a subspace of a vector space V. Suppose x, y € V are not in W. at if x e span (W,y) then y e span (W, x) (b) Give an example of a vector space V, a subspace W ≤ V, and vectors x, y € V such that xe span (W,y) but y span (W, x)