Elementary Matrices Generate the General Linear Group
Okay, so we can use elementary row operations to put any matrix into its (unique) reduced row echelon form. As we stated last time, this consists of building up a basis for the image of the transformation the matrix describes by walking through a basis for the domain space and either adding a new, independent basis vector or writing the image of a domain basis vector in terms of the existing image basis vectors.
So let’s say we’ve got a transformation in . Given a basis, we get an invertible matrix (which we’ll also call ). Then we can use elementary row operations to put this matrix into its reduced row echelon form. But now every basis vector gets sent to a vector that’s linearly independent of all the others, or else the transformation wouldn’t be invertible! That is, the reduced row echelon form of the matrix must be the identity matrix.
But remember that every one of our elementary row operations is the result of multiplying on the left by an elementary matrix. So we can take the matrices corresponding to the list of all the elementary row operations and write
which tells us that applying all these elementary row operations one after another leads us to the identity matrix. But this means that the product of all the elementary matrices on the right is . And since we can also apply this to the transformation , we can find a list of elementary matrices whose product is . That is, any invertible linear transformation can be written as the product of a finite list of elementary matrices, and thus the elementary matrices generate the general linear group.