Lec 19 - Subspaces and Spanning in a Vector Space

Wednesday, February 15, 2012

9:27 AM

 

Subspaces



 

This also happens in other vector spaces.

 

Definition





 

 Theorem: Subspace Test





 

Eg





 





 



 



 

Eg



 



 





 





 





 



 

Eg



 







 



 

 

Spanning

 

Definition





 

Definition



 

Eg





 







 

We get the following system of linear equations:









 

Put in a matrix: (Drop the 3rd equation)





 









 

Eg





 





 

We get the following system of linear equations:









 



 

The system is inconsistent.



 

 Theorem:



 

Proof: (by the subspace test)