When we convert a matrix into reduced row echelon form , the linearly independent vectors in the pivot columns form a unit vectors in the corresponding columns ? what is really happening here if I visualize it? we are performing row operations i.e operations on row vectors, so we are scaling down or up the row vectors by the same amount. But why this doesn't change the column vectors ?
Have a look at the Wikipedia entry on 'Guassian elimination'. The example on three simultaneous equations is the starting point. Then the idea of elementary row operations and what they do to the determinant of a matrix is next, in understanding how echelon form is useful in calculating the determinant of a matrix - but you have to be aware/happy that the determinant of an echelon form matrix is simply the product of its main diagaonal elements. Next is that reduced row echelon form is a way of getting an inverse matrix. There are other uses, but these three are reason enough for using the technique.