Let V be a vector space over a field F. Suppose that U and W are subspaces of V and define Z = {u + w | u ∈ U, w ∈ W}. a) Prove that the set Z is a subspace of V . b) Suppose that u1, . . . , ur span U and w1, . . . , ws span W. Find a spanning set for Z.

Let V be a vector space over a field F. Suppose that U and W are subspaces of V and define Z = {u + w | u ∈ U, w ∈ W}. a) Prove that the set Z is a subspace of V . b) Suppose that u1, . . . , ur span U and w1, . . . , ws span W. Find a spanning set for Z.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 24CM

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Question

Let V be a vector space over a field F. Suppose that U and W are subspaces of V and define
Z = {u + w | u ∈ U, w ∈ W}.
a) Prove that the set Z is a subspace of V .
b) Suppose that u1, . . . , ur
span U and w1, . . . , ws span W. Find a spanning set for Z.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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