Elementary matrices

Question

Is there a way to visualize the action of elementary matrices? (Or perhaps matrices in general). Perhaps someone could give an intuitive view of the effects of elementary row operations.

Actually I am trying to intuitively understand why these elementary matrices general $GL_n$.

Thanks.

Answer 1

Well,

1. When you act on geometric figures with the elementary matrix with a single non-one entry on the diagonal, you are stretching/shrinking the figure in the direction of that axis. If the entry is negative a reflection is also included. If the absolute value of that entry is $>1$, it's a stretch, if $<1$, you're shrinking things.
2. The elementary matrix swapping two rows gives the $n$-dimensional analogue of swapping the $xy$-coordinates on the plane. IOW, you are reflecting w.r.t to the line $x=y$.
3. The act of adding a scalar multiple of one row to another amounts to a shear. IOW, the mapping that turns a rectangle into a more general parallelogram by sliding the top. This action is sometimes seen in animated movies: the effect of slamming on the brakes of a moving car is visually accented by making the top of the car to briefly continue its motion, even after the wheels have already stopped.

To answer the question about why they generate the general linear group: There is no substitute to Gaussian elimination.