1. Vector Spaces Let (E, +,) be a vector space. Let (U, +, ) and (V, +,) be two vector subpaces of (E, +, ). Let U1. U2...., up} {V1. V2...., Vq) be two bases of U and V respectively. Prove that the following statement is true. The sum set U + V is a direct sum U O V if and only if u1, u2.- Up, V1, V2. are linearly independent. To perform the proof, invoke the pertinent definitions, theorems, lemmas, corollaries and propositions studied in this module.

1. Vector Spaces Let (E, +,) be a vector space. Let (U, +, ) and (V, +,) be two vector subpaces of (E, +, ). Let U1. U2...., up} {V1. V2...., Vq) be two bases of U and V respectively. Prove that the following statement is true. The sum set U + V is a direct sum U O V if and only if u1, u2.- Up, V1, V2. are linearly independent. To perform the proof, invoke the pertinent definitions, theorems, lemmas, corollaries and propositions studied in this module.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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