Lec 22 - Dimensions

Wednesday, February 29, 2012

9:30 AM

 

We can interpret a basis as:

1. A minimal spanning set of the vector space
2. A maximal linearly independent set of the vector space.

 

Definition





 

Eg









 

Eg





 

 Theorem:





1. 
2. 
3. 

 

Proof of 3:

















Q.E.D.

 

 Theorem:



Proof: A7

 

Eg





 







 

Eg





 



This gives a system of linear equations:







Row reduce the coefficient matrix:



Since rank of coefficient matrix equals the # of columns and the system Is consistent, the system has a unique solution.



 





 

 

 Coordinates

 

 







 

But we could also use another basis.

 

Eg







 

Definition





Note: Order of basis vectors matters.

 

Eg





 







 

Eg