Assume the vector spaces are finite for this problem. You don't need to do the problem for the vector spaces not being finite. Suppose V₁, ..., Vm, and W are vector spaces. Prove that L(V₁ × ··· × Vm, W) areisomorphic to L(V₁, W) × × L(Vm, W) vector spaces, where × represents the CartesianProduct. There is a shorter proof of this if one assumes that V₁, ..., Vm, and W are all finitedimensional, though one does not need to make this assumption to prove the statement.

Answered: Image /qna-images/answer/157c93c7-acbd-429f-99a0-50946fbc739f.jpg

Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) × XL(Vm, W) vector spaces, where X represents the Cartesian Product. There is a shorter proof of this if one assumes that V₁, ..., Vm, and W are all finite dimensional, though one does not need to make this assumption to prove the statement.

Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) × XL(Vm, W) vector spaces, where X represents the Cartesian Product. There is a shorter proof of this if one assumes that V₁, ..., Vm, and W are all finite dimensional, though one does not need to make this assumption to prove the statement.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 74E: Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors...

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