Recall the definition of Linear Independence: The set {V1,..., Vn} is linearly independent if, whenever a₁ V₁ + + an Vn = 0, it must be that = an = 0. a₁ = Suppose that {v2, V3} is linearly independent. Briefly describe what is wrong with the following "proof" that {V1, V2, V3} is linearly independent (where V₁ is some nonzero vector): Since (v2, V3} is linearly independent, a2 V2 + a3 V3 = 0. Then, if a₁ V₁ + a2V2 + a3 V3 = 0, we have that a₁ V₁ + 0 = 0 so that a₁ = 0. Therefore, {V1, V2, V3} is linearly independent.

Recall the definition of Linear Independence: The set {V1,..., Vn} is linearly independent if, whenever a₁ V₁ + + an Vn = 0, it must be that = an = 0. a₁ = Suppose that {v2, V3} is linearly independent. Briefly describe what is wrong with the following "proof" that {V1, V2, V3} is linearly independent (where V₁ is some nonzero vector): Since (v2, V3} is linearly independent, a2 V2 + a3 V3 = 0. Then, if a₁ V₁ + a2V2 + a3 V3 = 0, we have that a₁ V₁ + 0 = 0 so that a₁ = 0. Therefore, {V1, V2, V3} is linearly independent.

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 26EQ

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