Let V be a vector space over a field F with subspaces U and W. (b) Show that U + W = {u + w : u ∈ U, w ∈ W} is a subspace of V . (c) Is the subset (U + W) \ U ∩ W a subspace of V ? Explain your answer. (d) If B = {u1, . . . , un} is a basis for U and C = {w1, . . . , wm} is a basis for W, show that B ∪ C is a spanning set for U + W.

Let V be a vector space over a field F with subspaces U and W. (b) Show that U + W = {u + w : u ∈ U, w ∈ W} is a subspace of V . (c) Is the subset (U + W) \ U ∩ W a subspace of V ? Explain your answer. (d) If B = {u1, . . . , un} is a basis for U and C = {w1, . . . , wm} is a basis for W, show that B ∪ C is a spanning set for U + W.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.CR: Review Exercises

Problem 73CR

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