Vector Spaces: Basis for the column space

I need some clarification on finding the basis for the column space of a matrix. My steps are as follows

example: Let matrix A be 3 by 3 matrix
Step a.) take the row reduced echelon form of the matrix A
Step b.) identify linearly independent columns of rref A
Step c.) the corresponding colums of A matrix will be the column space fo A

My question is my books answer are different than what I have. I their is a standard form write the column basis.

Take the 3 by 3 matrix,

[4,20,31]
[6,-5,-6]
[2,-11,-16]

rref
~ [1,0,1/4]
[0,1,3/2]
[0,0,0]

using my step above I can see that the first two columns (left to right) are linearly independent.

so the column basis is,
{[4;6;2], [20,-5,-1]} ------> this my answer

my test book's answer is as follows,

{[1;0;-2/5], [0,1,3/5]}

Why are my answers different than my textbooks. If we are both correct how were my textbook answers derived.

i have no idea how the answers in your book were derived.

however, (4,6,2) = 4(1,0,-2/5) + 6(0,1,3/5), and

(20,-5,-11) = 20(1,0,-2/5) + (-5)(0,1,3/5), while

(1,0,-2/5) = (1/28)(4,6,2) + (3/70)(20,-5,-11) and

(0,1,3/5) = (1/7)(4,6,2) + (-1/35)(20,-5,-11)

so the two bases span the same space.

i suspect what your book did is use column reduction, rather than row reduction.