Unless otherwise stated, F is a field. V , W, and X are vector spaces over F. 2) Consider the field Fg. Let W = spanC Max2(F5). Is[1 2]E W? Your answer to this question does not need to be a full proof, butyou should show your work.

Answered: Image /qna-images/answer/f6b64c7e-ffc4-4e94-b8b7-e627a3538471.jpg

Answered: [3 2 [1 4] 2) Consider the field F5.…

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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Q1. Let V be a vector space over a field (F, +,.), let S={ vi, V2, V3 } be a linearly independent subset of V. Prove or disprove that the subset B={V1, Vr+v2, Vr+vz+v3 } is also a linearly independent subset of V.
6. Suppose that V is a vector space over a field F and that v1, v2, ..., Vn € V. For each statement below, decide if the statement is true. If you believe it is true give a proof; if not, give an example that contradicts the statement. a. If v1, v2, ..., Vn are linearly independent then vị, v2, . .. , Vn is a basis of V. b. If vi E (v2, vV3, . .. , Vn) then V1, V2, . .. , Vn are linearly dependent. c. If vi E (v1, v2, V3, . . . , Vn) then v1, v2, . . . , Vn are linearly dependent. d. If v1, v2, ..., Vn are linearly independent then (v1, v2, ... , Vn) 7 (v2, ..., Vn)
3.
. Define isomorphism of vector spaces. State its properties and give example for each property
Let V = R be the set of positive real numbers. We define addition and multiplication with standard numbers from the field R as follows: x +y = xy rx = x' r real number x,y EV Prove that V is the vector space for the above two operations
This picture shows a set of points in the plane R. Which of the following axioms of a vector space is NOT satisfied for this set? O a. Additive Inverses. O b. Additive Identity c. Closure under Addition O d. Commutativity in Addition
Please do part D and If the statement is true, you don't need to prove it. If the statement is not true, give a counterexample
Linear and quadratic vector fieldsa. For what values of a, b, c, and d is the fieldF = ⟨ax + by, cx + dy⟩ conservative?b. For what values of a, b, and c is the fieldF = ⟨ax2 - by2, cxy⟩ conservative?
Fix a field F. The direct sum of two F-vector spaces V and W is the F-vector space VW defined as follows. As a set, V W is equal to V x W, and addition and scalar multiplication on VW are defined component-wise: (v₁, W₁) +(v2, W₂) = (v₁+V2₂, W₁+W₂) and c(v, w) = (cv, cw). Problem 14. Prove that a function f: V → W is linear if and only if its graph¹ is a subspace of VW.
2. Let (V,+,) be a vector space over scalar field R. Suppose the set of vectors {v1, v2, v3, v4} is linearly independent and consider the following set, S={v1 2v2, V2, V3 V4 - V2, V4}. Determine if S is linearly independent. If not, remove as few vectors as you can so the resulting set is linearly independent and justify your choices.
2. Let n eN and suppose X is a set with n elements. Define V as the set of all functions f : X → C. The set V together operations + and · described below Vx e X, Vc e C and Vx € X (f + g)(x) = f(x) + g(x) (c. f)(x) = c · f(x) forms a vector space (don't verify this anymore). Find a basis of the vector space V.

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[3 2 [1 4] 2) Consider the field F5. Let W = span C Max2(F5). Is [1 2] 1 e W? Your answer to this question does not need to be a full proof, but you should show your work.