My linear algebra textbook claims that if the zero-vector is the only element in the null-space of a matrix, the following has to be true:

1) The column vectors of the matrix are linearly independent.

2) The reduced row echelon form of the coefficient matrix has to be an nxn identity matrix.

Point 1 is obvious to me, but I do not get why the matrix has to be a square matrix. If I have a, say, 3x2 matrix, and it's reduced row echelon form is:

10

01

00

Won't the column vectors in this case be linearly indepentent, and the only solution to the equation Ax=0 be the 2x1 zero-vector? Would appreciate any clarification!