EXAMPLE 1  Water Towers and Orthogonal Columns 

We will now consider the Water tower problem.  Our first goal is to model the changes in levels from minute k to k+1 and then to use Gram-Schmidt to redesign the problem so that the columns are linearly independent and orthogonal.  The tower system can be found under the title lecture 18 figure 1 

T1    (k+1)=(k) -(k) 

T2     = +  + +
T3    
T4    
We are really interested in space the difference between the levels at minute k+1 differs from the levels at minute k.  Let be the difference. 

T1  
T2  
T3    
T4  
 

We now look at the matrix representation of this system 

=
 

>	with(LinearAlgebra):

 

>	A:=<<-1,1/3,1/3,1/3>|<0,1,-1,0>|<0,1,0,-1>>;;

 

>	v1:=Column(A,1);

 

>	v2:=Column(A,2);

 

>	v3:=Column(A,3);

 

>	AG:=GramSchmidt([v1,v2,v3]);

 

>

 

>	v1a:=AG[1];

 

>	v2a:=AG[2];

 

>	v3a:=AG[3];

 

>	AGG:=<v1a|v2a|v3a>;

 

We will look at different orders of the vectors v1,v2,v3 when we use GramSchmidt to see if we get a set of orthogonal vectors that will be easier to work with.  

>	AG2:=GramSchmidt([v2,v3,v1]);

 

>	AG3:=GramSchmidt([v3,v2,v1]);

 

>	AG4:=GramSchmidt([v3,v1,v2]);

 

None of the above appear to improve our lot so we will design our water system using our new system with the coefficient matrix AGG 

>	AGG.<f[1](k),f[2](k),f[3](k)>;

 

>

 

The above columns are mutually orthogonal.  Making the coefficient matrix an orthogonal matrix.  i.e. a matrix where the columns are not only orthogonal but also each column of lenght one. (normalized)  Look at Lecture 18 figure 2 for the diagram.