Lec 2 - Span

Friday, January 06, 2012

9:27 AM

Recall: A set of vectors is closed under linear combinations if we can take a linear combination of vectors in the set and obtain a vector in the set.

 

Eg.



 

Eg.











 



 

Eg.

 







 

Note:





 

In general: Sets of vectors that are closed under linear combinations require only a small number of vectors to describe the entire set.

 

Definition





 



 

Eg.



 

Eg.





 

Eg.







 

Eg.





 

 How do we find the smallest # of vectors needed in a spanning set?

 

Theorem



 

Proof:





 





 











 







 



 




Q.E.D.

 

Eg.









 

 

 The vector equation of the plan is



 

Note: To find a smallest spanning set, we need to identify when one vector the set is a linear combination of the remaining vectors in the set.

 

 Linear Dependence

 

Definition

A set of vectors is linearly dependent if one vector in the set is a linear combination of the remaining vectors.

Otherwise, the set of vectors is linearly independent.

 

Eg.