]> 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