MDL

A sweeping example: the 3x3 matrix A with entries 123456789.

We use the 3x6 matrix as in the algorithm in §2.2 for finding the inverse of A, which doesn't exist.

Solving Ax=y we run into a consistency condition for y.

From that condition we obtain a nice basis for the range R(A) of A.

And R(A) is really the column space of A.

The row sweeping method used is described in Lay's Section 1.2.

The theorem that eventually follows are Theorem 13 and 14 in Sections 2.8 and 2.9.

The nice basis obtained is the column sweep of A.

Which must be the row sweep of the transpose of A.

The last 2 observations are mine.

NB This A and its transpose have the same null spaces and ranges. This is not always the case!

As a square matrix A was already special because N(A) was more than just {0}.

NB is similar/equivalent to a 3x3 matrix B which is really a 2x2 matrix if you think of it.