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3.5 Linear Dependence and Independence

A linear dependency among vectors v 1 ⟶ to v k ⟶ is an equation, c 1 v 1 ⟶ + ⋯ + c k v k ⟶ = 0 ⟶ , or ∑ j c j v j ⟶ = 0 ⟶ in which some of the c's ar not 0. A set of vectors is said to be linearly independent if there is no linear dependence among them, and linearly dependent if there is one or more linear dependence.

Example: suppose v 1 ⟶ = i ^ + j ^ ; v 2 ⟶ = 2 i ^ ; v 3 ⟶ = 3 j ^ .

Then v 1 ⟶ , v 2 ⟶ and v 3 ⟶ are linearly dependent because there is the relation

6 v 1 ⟶ = 3 v 2 ⟶ + 2 v 3 ⟶ , or 6 v 1 ⟶ − 3 v 2 ⟶ − 2 v 3 ⟶ = 0 ⟶

Exercise 3.11 Prove: any k + 1 k-vectors are linearly dependent. (You can do it by using mathematical induction.) (If you are not familiar with mathematical induction read this solution and become familiar with it!) Solution

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