Question

Does row independence imply column independence?

Answer 1

If the rows of AA are linearly independent, then the result of doing row-reduction to AA is the identity matrix, so the only solution of Av=0Av=0 is v=0v=0.

If the columns of AA are linearly dependent, say,

a1c1+a2c2+⋯+ancn=0

a1c1+a2c2+⋯+ancn=0

where the cici are the columns and the aiai are not all zero, then Av=0Av=0 where

v=(a1,a2,…,an)≠0

v=(a1,a2,…,an)≠0

So, if the columns are dependent, then so are the rows.

Now apply the same argument to the transpose of AA to conclude that if the rows of AA are dependent then so are the columns.