What is the significance of nontrivial solutions in linear algebraic equations?

[firecool]

finding out non trivial solution of two linear algebraic equations means we have to equate the determinant of the coefficients to zero which in turn means the slope of the two lines are same and they lie on each other. so what is the use of two equations for this? can we get the points or solution from just one equation? and it basically implies that both the lines are same. I'm getting confused here over the significance of the nontrivial solutions. i came across this in the derivation of frequencies of systems with two degress of freedom.

 

[HallsofIvy]

I wonderred why 77 people had looked at this thread but none had responded. Now, I see. It is almost impossible to understand what you are saying. When looking for "non-trivial solutions of two linear algebraic equations" we do NOT "equate the determinant of the coefficients to zero". We do that only when working with homogeneous equations. I.e. something like ax+ by= 0, cx+ dy= 0. If the determinant ad- bc is not 0, then the only solution is x= y= 0. But if ad- bc= 0 then, yes, the two equations are equivalent and there exist an infinite number of solutions: choose x to be anything and y= -(a/b)x which is the same as y= -(c/d)x. You are right that you don't need both equations- but you don't know that until you find that the determinant is 0. And you need both equations to do that.

Typically, such a problem occurs when you have other conditions as well. If your solutions were "trivial" you would not be able to satisfy those additional conditions. So you start with two equations and, typically, see that some parameters must have a certain value so that your equations are the same and you do have non-trivial solutions and can satisfy the other conditions.

 

[firecool]

thanks! :) i think i get it..