Hey PF, I'm having trouble seeing the bigger picture here. Take matrix A and matrix B. If B can be obtained from A by elementary row operations then the two matrices are row equivalent. The only explanation my book gives is that since B was obtained by elementary row operations, (scalar multiplication and vector addition) that the row vectors of B can be written in terms of the row vectors in A. I follow this sort of, I at least see the 'intuitiveness' behind it but I feel like there is some piece of the puzzle missing in my brain that's not letting it click completely. Anyone want to drop some knowledge on me? And while you're at it answer me this: why aren't matrices A and B guaranteed to be column equivalent? What is the difference between rows and columns? If you get matrix A from matrix B by only elementary-column operations (is there such a thing?) then that would mean they are guaranteed to be column equivalent?