-Problem 7. Let W = {(a1, a2, a3) € R³ : 5a² − 3a² + 6a² = 0}. Is W a subspace of R³ underthe operations of addition and scalar multiplication defined on R³? Justify your answers.Problem. Prove that there is a field F such that W = {(a₁, a2, a3) € F³: a₁ = a3 + 2} is asubspace of F3.Remark. As discussed in class F = R cannot be a solution to the previous question.Problem. Let W₁ and W₂ be subspaces of a vector space V. Prove that W₁ UW₂ is a subspaceof V if and only if W₁ C W₂ and W₂ C W₁.2

Answered: Image /qna-images/answer/85267884-c520-45fc-9058-54801091f802.jpg

Answered: Problem 7. Let W = {(a1, a2, a3) € R³ :…

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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(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
II. Solve all problems completely. Let V = {x : x € R*} (set of positive real numbers) with the operations 1. x +y x+y and r•x= x". 2 = Is vector addition associative? Show your solution. b. What is the "zero" element in V? Is it unique? Explain your answer. Is V closed under scalar multiplication? Explain your answer. d. Is the distributive law of vector multiplication over addition (M2) satisfied? Show your solution. а. С.
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Problem 7. Let W = {(a1, a2, a3) € R³ : 5a² − 3a² + 6a² = 0}. Is W a subspace of R³ under the operations of addition and scalar multiplication defined on R³? Justify your answers.