multiplying a linear system by an invertible diagonal matrix let $Ax = b$ be a linear n by n system
if we multiply this system by a non-singular diagonal $D$ I can say that the new system still has the same solution as the previous one , right ?
I ran some tests on matlab and noticed that the condition number stays all the time the same but our teacher claims that by this method we can make the condition number smaller of the system and he didn't explain why
can someone please clarify or give me an example where it is true ? thanks !

Regarding the solution space question, multiplying both sides of this equation by a non-singular diagonal matrix corresponds to multiplying each equation in a system of linear equations by some non-zero integer. Looking at it that way, it is clear that the solution space stays the same. As an example, the system
$$x+y=7$$
$$x+2y=10$$
has the same solutions as 
$$2x+2y=14$$
$$3x+6y=30$$
