Are column operations legal in matrices also? In linear algebra we have been talking a lot about the three elementary row operations. What I don't understand is why we can't multiply by any column by a constant? Since a matrix is really just a grouping of column vectors, shouldn't we be able to multiply a whole column by a constant but maintain the same set of solutions for the original and resulting matrices?

i think that the reason which causes "elementary" column operations incorrect is because of our rules toward the form of linear equations. That is, we define linear equations to be written as what we get used to writing now! You can try to write down theses equations straight and apply column operations to them, and see what happens.
