]> 3.5 Linear Dependence and Independence

## 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