Creating a linear combination in terms of the free variable in MATLAB

In MATLAB I want to create a code that gets an Augmented matrix (Concatenating A and B) and produce the linear combination of the system in terms of the free variables. The free variables will be given the variables t1, t2, ..., tn for the nth free variables. I want to fix the code below so that the inputs match the outputs. The code should be able to output the linear combination of the system in terms of the free variables regardless of the size of the matrix, for example, a 1 x 10, 5 x 5, or 7 x 3. Look at the 2 test cases below and make sure that the code could output the linear combination as a whole with its column matrices and free variables.

Code:

A = [2 3 4 56 6 8; 1 2 4 5 6 7; 3 6 12 15 18 21; 4 6 8 112 12 16];
B = [9; 8; 24; 18];
Aug = [A B];
RREF = rref(Aug);

[m, n] = size(RREF);

free_variables = find(all(RREF(:,1:n-1)==0, 2));

for i=1:n-1
    if ~any(RREF(:,i))
        fprintf('x%d = ', i);
        fprintf('%s', '0 ');
        for j=1:length(free_variables)
            fprintf('+ %dt%d ', RREF(free_variables(j), n), free_variables(j));
        end
        fprintf('\n');
    else
        [~, j] = max(RREF(:,i));
        fprintf('x%d = ', i);
        fprintf('%dt%d ', RREF(j, n), j);
        for k=1:n-1
            if k~=i && RREF(j,k)~=0
                fprintf('%+dt%d ', -1*RREF(j,k), k);
            end
        end
        for j=1:length(free_variables)
            fprintf('%+dt%d ', RREF(free_variables(j), n), free_variables(j));
        end
        fprintf('\n');
    end
end

Test Cases

Input:

Output:

x1= 4t1 - 97t2 +6t3 + 5t4 +6t5

x2= -4t1 + 46t2 - 6t3 - 6t4 - 7t5

x3= t1

x4= t2

x5= t3

x6= t4

x7= t5

Input:

Output:

x1= -4/3t1 - 5/3t2 - 1/3t3

x2= t1

x3= t2

[Matt J]

Since you already have code, it's hard to know what your question is.

[Teoman]

Im trying to get the row opertaions needed to get to the rref of a matrix. The code above does not give needed output

[Dyuman Joshi]

What is the output you want?

Answer 1

[Rohit]

I understand that given a system of equations, you want to convert them in matrix form and then represent it in terms of free variables.

Based on your expected output, it looks like matrix B is another matrix you are concatenating and not RHS of equations.

So, in your current code, there seems an issue in finding free variables and using them correctly to get relevant equations.

I have written a basic code which gives your expected output, and you can modify it as per your requirements.

A = [2 3 4 56 6 8; 1 2 4 5 6 7; 3 6 12 15 18 21; 4 6 8 112 12 16];
B = [9; 8; 24; 18];
% A= [3,4,5 ; 1.5,2,2.5];
% B= [1;1/2];
Aug = [A B];
%calculating pivot and free variables
[RREF,pivot_variables] = rref(Aug);
[m, n] = size(RREF);
free_variables =setdiff(1:size(Aug,2),pivot_variables);

for j = 1:n
    if ismember(j, pivot_variables)
        fprintf('x%d = ', j);
        for k = 1:n
            if k ~= j && ismember(k,free_variables) %writing pivot variable as combination of free variables
                fprintf(' %+d*t%d', -RREF(find(RREF(:,j),1),k), k-length(pivot_variables));
            end
        end
    else
        % print the free variable
        fprintf('x%d =', j);
        for k = 1:length(free_variables)
            if k == j-length(pivot_variables)
                fprintf(' t%d', k);
            end
        end
    end
    fprintf('\n');
end

x1 = 

 +4*t1 -97*t2 +6*t3 +5*t4 +6*t5

x2 = 

 -4*t1 +46*t2 -6*t3 -6*t4 -7*t5

x3 =

 t1

x4 =

 t2

x5 =

 t3

x6 =

 t4

x7 =

 t5