Let V be an n-dimensional vector space over F and B = {b₁,b2,...,bk} be a linearlyindependent set of vectors in V where n > k.Suppose W = span(B). Prove that every vector a ¤ V can be expressed as α=α₁+ α₂where a ₁ € W and a 2 W. Show that a ₁ is unique but a 2 is not unique.

Introduction: A set B of vectors in a vector space V is called a basis if every element of V may…

Answered: Let V be an n-dimensional vector space…

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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Let V be an n-dimensional vector space over F and B = {b₁,b2,...,b} be a linearly independent set of vectors in V where n > k. Suppose W = span(B). Prove that every vector a E V can be expressed as α = α₁ + α₂ where a ₁ EW and a 2 W. Show that a ₁ is unique but a 2 is not unique.