# identifying independent columns when solving a homogeneous set of equations i.e. Ax = 0

Given a matrix say $A$ with columns $a_1, a_2, ..., a_n$, and say it is reduced to a reduced-row echelon form with the entry 1 in the pivot say denoted the reduced-row echelon matrix as R with the column being $r_1, r_2, ..., r_n$. I was reading that the columns that are independent in $A$ is the same column as $R$. i..e. if say the column $r_3, r_7, r_8$ is identified as independent, then so is the $a_2,a_7,a_8$ columns in $A$.

When we try to identify whether a set of column in the reduced-row echelon matrix $R$ is independent or not, I know that if there is a leading 1 in the column, then it is a pivot column, so it will be independent with other columns that also has a leading 1 in the column.

But my question is can we identify EXACTLY which set of columns is independent in $R$ by looking at the entries of the element in the vector $x$ when solving the system of equation $Rx=0$ ? say $R$ is a m x n matrix, with n columns, therefore $x = (x_1, x_2,...,x_n)$. Now say consider all the solution $x$ to $Rx=0$, if certain entries say $x_2, x_4, x_{11}$ ALWAYS equal to zero in solving $Rx=0$, i.e. $x = (x_1, 0, x_3, 0, ..., x_{10}, 0, x_{12}, ...,x_n)$ , then can we say the column $r_2, r_4, r_{11}$ are linearly independent with each other? and also these 3 columns are linear independent with the rest of the other columns in R?
and then say the corresponding columns in A, i.e. $a_2, a_4, a_{11}$ are linearly independent with each other?

My main question is can we look at the entries in the vector $x$ to identify which corresponding columns in R is independent?

Say for all the solution to the $Rx=0$ system, for simplicity say R is a 5 by 7 matrix only, and I ALWAYS have this:

$x_1 = ( 3, 0 , 0, 2, 1, 9, 0)$

$x_2 = ( 5, 0 , 0, 3, 2, 3, 0)$

$x_3 = ( 3, 0 , 0, 1, 9, 11, 0)$

$x_4 = ( 2, 0 , 0, 2, 0, 9, 0)$

$x_5 = ( 5, 0 , 0, 3, 2, 3, 0)$

$x_6 = (4, 0 , 0, -1, 0, 11, 0)$

$x_7 = ( -8, 0 , 0, 1, 9, 10, 0)$

$x_8 = ( 1, 0 , 0, -2, -5, 2, 0)$

etc.... i.e. the Second entry, Third entry, and Last entry are always ZERO at the same time, can I say the second, third, and last column in the matrix $R$ are linearly independent with each other?, and also linearly independent with the rest of the columns in $R$?