Show that there do not exist scalars C₁, C₂, and C3such thatc₁(1,0, 1,0) + c₂ (1, 0, 2, 1) + c3(2,0, 1, 2) = (1, -2, 2, 3)

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Answered: Show that there do not exist scalars…

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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Let v = {(; Define addition and scalar multiplication as follows. 1 b +e kb kc verify the following axioms: a) (u + v) + w = u + (v + w) b) u + (-u) = –u + u = 0 c) c. (u + v) = (c.u) + (c.v) Note: u, v, w, are vectors and c and d are scalars
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43. Linear combination Let u = (1, 2, 1), v = (1,-1,-1), w = (1, 1,-1), and z = (2, -3,-4). Find scalars a, b, and c such that z = au + bv + cw.
Suppose that ū, ī and w are vectors in a real inner product space such that (7, w) = 16, (T, w) = 5, d(2ū, v) = v27 and ||W|| = V27. Show that 27 - 0 - w = 0.
3. (12) Let V = R². Define vector addition and scalar multiplication as follows: (u1, uz)O(v1, v2) = (u1V1, U2V2) and c(u1,u2) = (u,C, u2^), c > 0 (a) Does V have an additive identity? If so, what would it be? (b) Does the distributive axiom c(uOv) = cu@cv hold? (Show why or why not). 4
(u1 + v1, U2 + v2) and au = (au1, 2), respectively. Let V = {(T,y)r, Y ER}, with addition and scalar multiplication defined as u + v = (u1, U2) and v = (v1, V2) and scalars a. Which of the following is true? where u = O A. k(lu) = (kl)u and 1u = u for all E V O B. (0,0) is the zero of V and (kl)u = k(lu) for all scalars k, l and u E V || O C. Addition is associative and k(u + v) ku + kv for all u, v E V and for all scalars k O D. 3(u + v) : 3u + 3v and the negative of (-2, 3) is (2, –3) O E. k(u + v) = ku + kv for u, v E V and scalar k, and addition is commutative Reset Selection

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Show that there do not exist scalars c₁, c₂, and C3 such that c₁(1,0, 1,0) + c2 (1, 0, 2, 1) + c3(2,0, 1, 2) = (1, -2, 2, 3)