Problem 1 (Changing the generating set of a span) This problem has two parts: (a) Show that if v is a linear combination of v1, ..., Vm and each vi is a linear combination of w1, ..., Wn, then v is a linear combination of W1, ..., wn. (b) Suppose v1, ..., Vm are vectors that span a vector space V. Let w₁, ..., wn Є V be any other elements. Suppose that each v¿ can be written as a linear combination of w₁, ..., Wn. Prove that the vectors w₁, ..., Wn also span V. Challenge: If it so happens that m < n, can you also prove that out of the n vectors w₁,..., Wn, you can pick m of them that will still span V?

Problem 1 (Changing the generating set of a span) This problem has two parts: (a) Show that if v is a linear combination of v1, ..., Vm and each vi is a linear combination of w1, ..., Wn, then v is a linear combination of W1, ..., wn. (b) Suppose v1, ..., Vm are vectors that span a vector space V. Let w₁, ..., wn Є V be any other elements. Suppose that each v¿ can be written as a linear combination of w₁, ..., Wn. Prove that the vectors w₁, ..., Wn also span V. Challenge: If it so happens that m < n, can you also prove that out of the n vectors w₁,..., Wn, you can pick m of them that will still span V?

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.3: Spanning Sets And Linear Independence

Problem 21EQ

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