My professor gave us this definition for a system of equations in echelon form:     A system of m linear equations in n variables is called an echelon system if         m ≤ n.   Every variable is the leading variable of at most one equation.   Every leading variable is to the left of the leading variables of all lower equations.   Every equation has a leading variable.      He stressed the importance of this form, but didn't actually give us a reason why it was important. Why is this form so important? It seems kinda random.
My prof gave us this definition of an Echelon system:     A system of m linear equations in n variables is called an echelon system if         m ≤ n.   Every variable is the leading variable of at most one equation.   Every leading variable is to the left of the leading variables of all lower equations.   Every equation has a leading variable.      I don't understand this definition at all.  Q1)If you have this system of equations:  $ax_1 + bx_2 + cx_3 = z$  $dx_1 + ex_2 + fx_3 = z$  its not an echelon system because of multiple equations having the same variable as the leading variable and therefore violating rule #2.  But if you switch around variables and rewrite the equation as:  $ax_1 + bx_2 + cx_3 = z$  $ex_2 + dx_1 + fx_3 = z$  It becomes an echelon system! Why would the way you write a system of equations affect its definition?  Q2)Why would point number three matter? Can't you just move around the equations so that they are in order of variables?  Q3)Why would point number four matter? If you have a equation in the system that doesn't have a leading variable can't you just disregard it? For example if you have the equation $0x = 0$ in the equation can't you just ignore it to have a echelon system since it doesn't affect the solution of the system?
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