In my linear algebra text, there is a theorem that says row operations and column operations don't affect the dimension of the column space and row space of a matrix. If I were solving a system of linear equations, will collumn operations mess up the system. I am confused about what this theorem means?
No, this isn't how you are supposed to think about it.
If you sole purpose is to solve a linear system in the form
, then column operations are basically useless, because you are not interested in solving for values of e.g.
, but value of
Similarly, if your sole purpose is to solve a linear system of the form
, then row operations are basically useless for a similar reason.
Instead, this is a theorem because it asserts two different ways of defining rank (i.e. dimension of row space, dimension of column space) are actually the same thing, so we can speak of the rank of a matrix without specifying whether we mean the dimension of row space, or the dimension of column space. Fom the proof of this we get, for example, the rank-nullity formula.