Question

Recall the methods for finding bases of the row and column space of a matrix A which were shown in lectures. To find a basis for the row space we row reduce and then take the non zero rows in reduced row echelon form, and to find a basis for the column space we row reduce to find the pivot columns and then take the corresponding columns from the original matrix. In this question we consider what happens if we mix up these techniques.

(a) Suppose that you take the pivot columns from the row reduced matrix, will this in general give a basis for the column space?

(b) Suppose that you identify the non-zero rows of the row reduced matrix, but then take the corresponding rows of the original ma- trix, will this in general give a basis for the row space?

In each case you must give arguments and specific examples where applicable to support your answers.

Answer 1