hi there,

I have been looking at matrices, and so far I understand that if we have a system of linear equations, then a matrix is used to present the coefficients of the system in a 2-dimensional grid format.

I also understand that the 'solution' to a system of linear equations, will be in the form of a coordinate (or set of coordinates) at the point(s) where the lines/planes cross each other... Basically solving simultaneous equations. (though the method of solving a system using matrices comesfurther onin the text I'm reading; Gaussian elimination I think its called?)

What I don't understand is when it says something like:

"Solve the system of linear equations over Z_p"

I have briefly looked at "set theory" and I believe that Z_p is supposed to represent a certain "set" of numbers. I also gather that in this case the coefficients of the system must be elements (numbers) within this set and nothing else.

So basically the above quotation can be thought of as: "find the coordinate(s) of any points of intersection of the lines and planes involved when the coefficients of their equations are numbers belonging to the set, Z-p". Would that be accurate to say?

2 more questions I have following that are:

1.) In terms of application, why exactly would we need to say that the coefficients of the system must belong to a particular set and nothing else? I mean why can't we just use any number?

2.) Amongst the jargon of all the texts I've read on this subject, there is always the mention of a 'field' as well. This term seems to be used interchangeably (or very closely linked) with the term 'set', so what exactly is the difference (if any) between the two?

I hope someone here can answer these questions,

thanks,

bit4bit