I've studied linear algebra before, however, I wanted to come back to the foundations and understand it again from the beginning.
I was looking the following inoffensive linear equations:
$$ x – 2y = 1 $$
$$ 3x + 2y = 11 $$
and after elimination one has:
$$ x – 2y = 1 $$
$$ 8y = 8 $$
the claim is that both systems of equations have the same solution. Geometrically:
if you've never seen this before it nearly seems like magic! We managed to combine different equations and still retain that the solutions is preserved. The new equations seems to be completely different and the lines that we care about don't even have same gradients. Basically, at first glance, the problem seems like it dramatically changed!
So even though, the system seems to have changed a lot, in reality, it didn't change that much since they still intersect at the same point.
My question is, what is the intuition to why manipulating system of equations in this way and combining them retains the original solution.
The intuition/justification that I used to think of was that, if we combine equations, in principal, the "total" information that we had at the beginning of the system is preserved as long we are combining different equations and we don't discard one any of them. Basically, combining two equations implicitly retains the information that we had about the old equation. However, we can "forget" about the old form of the new equation because that information is preserved even though the equation changed. i.e. its ok to combine equation 1 and 2 to form 2' and discard 2, since 2' AND 1 contains all the information of the original system.
This is sort of the intuition that I use but I wasn't sure if that was a good way to think about it or if people maybe had better intuitions or justification to why elimination worked.
Best Answer
I will likely be proven wrong by another answer, but I don't think that you can get a geometrical explanation. Algebraically, the situation is very clear: you have a system $Ax=b$, and when you perform row reduction, you are multiplying on the left by an invertible (elementary) matrix. So the system becomes $EAx=Eb$, with the same solution. Multiplying by an invertible matrix does not have an easy geometrical interpretation, as far as I can tell.