# Basis for a column space of a matrix

Say there is a matrix A with 77 columns and the rank of AA is 55. Columns 1,3,4,51, 3,4, 5, and 77 are linearly independent (not necessarily pivot columns). Is it true or false that these linearly independent columns form a basis for the column space of A?

I think that this statement is true. My reasoning is that if these 5 columns are linearly independent, then columns 2 and 6 can be written as linear combinations of these linearly independent columns. Therefore, since those 5 columns can generate columns 22 and 66, they span the column space of A. So, since they are in and span the column space of AA, and they are linearly independent, that fulfills the definition of a basis for a subspace, and that must mean those 55 columns do form a basis for the column space of AA.

Is this statement true or false? And is the reasoning alright? Any insight would be awesome.
(Note: I am currently taking linear algebra. I’m not a math major or anything so bear with me if the explanation is somewhat mediocre.)

Thank you

=================

=================