Ok, so I am taking my first course in linear algebra, and even though I am not a math major (physics major actually), I can't help but wish my teacher and text were more rigorous. So let me start by telling you all the problem I am having:

(First question) My book states the following rank theorem: a system of m eqns in n variable, there will be exactly n-r parameters. r being rank of augmented matrix. But then I begun thinking why, the proof my book gave is a joke, just logical words, and I can't seem to understand any proofs online. So I thought, hey can't you have a matrix with for say a system of 5 equations in 5 variables and in the augmented matrix have only say, 3 leading ones, and instead of the fourth non leading variable, have a column of all zeros, and a zero row at the bottom. I hope you understand what I am saying. Like through applying the Gaussian algorithm that's what you end up with. Here there surely would not be a n-r parameters.

Now come my next question, which is somewhat related. Can you perform elementary many linear operations simultaneously. For example given augmented matrix (here brackets represent a row, and column number increases left to right):

[(1,1,2,3),(-2,-1,0,-4),(4,2,4,7)) can I simply ---> R1 - 1/2R2 and do R2-2R1 --> end up with matrix with an entire column complete cleared out and zeros in it's place.

I know this is probably in violation of some basic properties, so I would like someone to explain why this is so. Why you can't perform multiple elementary row operations at once. Would you still not end up with an equivalent system, since a solution should work for each equation in the original, so can't you keep adding original equations and stuff all at once. Also if someone could help me with my first question.

Ahh, this is killing me.