Find a basis for ...

Finding a basis for a row space of A by row reductions:

The position of the pivots of Ab gives the position of the rows of A which form a basis for the row space of A.

Find a basis for the orthogonal complement of A:

By calculating the free variables of the system Ab we will obtain this basis.

Finding a basis for a Column Space by Row Reduction ::

The position of the pivots of Ab gives the position of the columns of A which form a basis for the column space of A.

Find a basis for the orthogonal complement of AT:

The free vectors of the system BTb will be a basis for the orthogonal complement of AT, with B a basis of the column space of A.