What's the underlying thing about reducing a matrix to reduced echelon form that solves so many things for us?
Some determinations that can be from reducing to reduced echelon form:
- Testing linear dependency
- Seeing if a matrix has an inverse
- Solving a system of linear equations
- Finding the number of free parameters in a system of linear equations
- Finding row and column rank
I know why it can determine each individual thing, but there seems to be some underlying essence behind it all. It seems like a very convenient algorithm, so there must be something about it I'm missing that explains why it can explain so many things about a matrix. My guess is that it reduces the matrix down to its basis which may determine the rest by analyzing the basis, but other than that, not sure. What gives?
ADDENDUM:
I've been told in a comment that "reducing to RREF reveals the basis of the row-space". How can a matrix have a different row space and column space simultaneously? What does that mean about matrices, exactly?