If a set of row vectors are linearly independent and the matrix of those row vectors is a square matrix, does it means the same of column vectors?

Question

If a set of row vectors are linearly independent and the matrix of those row vectors is a square matrix, does it means the set of column vectors are linearly independent and vice versa?

I already know that if A is not a square matrix, then there're some examples which the row vectors are linearly independent and the column vectors are not(linearly dependent), but I do not sure if the statement is true when A is a square matrix.

Can someone help me to figure out this questions or give me some counter examples or algebraic proof?

Answer 1

Yup, the columns are also linearly independent and vice versa!

This is a special case of the fact that, for all matrices, column rank and row rank are equal (see many explanations for this fact here).

If the row vectors are linearly independent, then $A$ has full row rank. Therefore, $A$ has full column rank. Therefore, the columns of $A$ are linearly independent!