Let V = Mnxn(F). (a) Suppose that {M1,..., Mt} C Mnxn(F) is a spanning set. Prove that {M{,.…, M£} is also a spanning set. (Hint: Express At as a linear combination of the M;'s and use transpose carefully.) (b) Prove that if B = {M11, M12, -.. , Mnn} is any basis of V, then B' = {M, M, ..., Min} is also a basis of V.

Let V = Mnxn(F). (a) Suppose that {M1,..., Mt} C Mnxn(F) is a spanning set. Prove that {M{,.…, M£} is also a spanning set. (Hint: Express At as a linear combination of the M;'s and use transpose carefully.) (b) Prove that if B = {M11, M12, -.. , Mnn} is any basis of V, then B' = {M, M, ..., Min} is also a basis of V.

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 4CM: Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4,...

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