An Approach to Matrix & Linear Algebra

by Frank C. Fung ( 1st published in November, 2004. )

Top Of Page

Section 1 : Linear Algebra for 3 variables

Topic I - Introduction & Summary for the Section :

In this section, we shall look at a set of { 3 Linear Equations } and attempt to find the general solution.

go to Top Of Page

Topic II - Setting up the { 3 Linear Equations } :

Let us first set up a set of { 3 Linear Equations } for the variables 'X' , 'Y' , & 'Z' , as follows :

where :

• the values of 'A' , 'B' , 'C' , 'D' , 'E' , 'F' , 'G' , 'H' & 'J' are fixed,

• the values of '' , '' , & '' are given.

go to Top Of Page

Topic III - Setting up the format for the solution :

Let us suppose that we were able to find :

• a set of three (3) values, '' , '' , & '' , respectively, such that :

  

• a set of three (3) values, '' , '' , & '', respectively, such that :

  

• a set of three (3) values, '' , '' , & '', respectively, such that :

  

Then, for the original set of { 3 Linear Equations } , i.e. :

the solution is given by :

or alternately, in the { matrix notations } format :

go to Top Of Page

Topic IV - Finding the solution :

Let us now set up the value [ VOD ] , the value of the { Determinant } for the original matrix.

And this value is given by :

Using a numerical method procedure, the { Runge Katta Method } , for a { 3 x 3 matrix } , with details in Appendix A , we were able to determine that :

We then have :

go to Top Of Page

Topic V - Setting up the { Linear Mapping } relation :

We note here that the equation :

dictates that :

• for every set of values of { , , } ,

• there is a corresponding set of values of { X , Y , Z } ,

  so that there is now a { linear mapping } relation.

In the next section, we shall be taking a closer look at this { linear mapping } relation.

go to Top Of Page

go to the next Section : Section 2 - Linear Mapping Considerations

return to FCF's Matrix & Linear Algebra HomePage

original dated 2004-11-15