Matrix solution for k>n, k<=n, and k=n equations and variables

[nhrock3]

i can't think theoretically here.
there is a homogeneos equation system which has k equations and n variables.

what type of solution we have if for k>n?

what type of solution we have if for k<=n?

first of all i can't deside if the equatins oare independent.

if k>n then we have endless solutions because of the free variables.

if k<=n then we will have one of the equations which will turn to a line of zeros

so i don't know what is the solution of this

 

[Char. Limit]

Well, I'd actually divide it into three situations...

k>n, k=n, and k<n.

If k<n, that is, there are less equations than variables, then you have free variables and endless solutions.

If k>n, that is, there are more equations than variables, then it's possible that you have no solution, one solution, or infinite solutions, depending on linear independency and some other factor that I can't name.

If k=n and all of the equations are linearly independent, then there should be 1 solution.