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.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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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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