Finding the columns that contain 1 and 0s(Basic Columns) in MATLAB

I am attempting to create a piece of code that can identify the basic and non basic columns from a matrix. The basic columns are going to be the ones that contain only 1s and 0s, and the arrangement of the 1s must be different according to the rank of the matrix. For example, if the matrix has a rank of 2 (variable RANK), then there will be a column with a 1 in the 0th index and another column with a 1 at the 1st index. The rest of the values other than 1 will be full of 0s. Hence the number of basic columns must be equivalent to the RANK variable. The columns that are not basic will just be the non basic columns. I am unsure of the best method to code such a thing that will accurately showcase the basic and non basic columns. Note that there could be 0 basic columns or 0 non basic columns. Please get the indexes starting from 1 to identify the basic and non basic columns. In the outputs for the test cases X1 is the first column of the RREF matrix from the very left whilst X3 is the 3rd column.

Code:

A = [3 2 4;3 3 6];
RREF = rref(A) 
RANK = rank(A)

Test Cases

Output:

Basic Columns: {X1, X2}

Non Basic Columns: {X3, X4, X5}

Output:

Basic Columns: {X1}

Non Basic Columns: {X2, X3, X4}

Input:

Output:

Basic Columns: {X1, X2, X3}

Non Basic Columns: {}

Answer 1

[Matt J]

Well, I definiitely don't think you should be using rref, which is a very primitive command. Instead , you should use this FEX download,

https://www.mathworks.com/matlabcentral/fileexchange/77437-extract-linearly-independent-subset-of-matrix-columns?s_tid=srchtitle

Note that the choice of basic columns will not be unique, in general.

A = [3 2 4;3 3 6]

A = 2×3

3 2 4 3 3 6

[X, I]=licols(A);

A_reduced=X\A;
A_reduced(end+1:size(A,1),:)=0

A_reduced = 2×3

1.0000 0 0 0 0.5000 1.0000

basiccols=A_reduced(:,I)

basiccols = 2×2

1 0 0 1

nonbasiccols=A_reduced(:,setdiff(1:size(A,2),I))

nonbasiccols = 2×1

0 0.5000