Lec 3  - Basis & Higher Dimensions

Monday, January 09, 2012

9:24 AM

 

 Q: How do we find the smallest spanning set?

 





 

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.

 

 

 Is there a nice mathematical expression to define linear dependence?

 



 





 







 



 

 



 

Better Definition







 

 

 

 

Proof:

 







Q.E.D.

 

 

Eg.




Consider the equation





 

We have 3 equations and 3 unknowns:

 



 



 

Sub into equation 1:



 



 

Our set is linearly dependent.

 

 

 Note: In order for a spanning set to be as small as possible, it must be linearly independent.

 

Definition



 



 

Ex.

 



 



 

Eg.



 

 

 Surfaces in Higher Dimensions

 

Definition





 



 





 

 

 Subspaces

 

For a set of vectors to be closed under linear combinations, we must be able to apply the operations of vector addition and scalar multiplication.

 

Definition





 



 

Definition