Are Column Vectors in AB Unchanged if Linearly Independent?

[only_huce]

If you have a 3x3 matrix AB of rank 2, and you are asked to find two linearly independent vectors in the column space of A, are those the same vectors in the column space of AB? I figure it is since if the vectors are linearly independent they wouldn't change. Am I right?

 

[adriank]

That's right. If x is in the column space of AB, then x = ABy for some vector y. But since x = A(By), x is in the column space of A. Thus, if you can find two linearly independent vectors in col AB, you've found two linearly independent vectors in col A.

Be careful, as the opposite isn't true: it's not necessarily true that if x is in the column space of A, then it's in the column space of AB.