Rank of a matrix is the no of pivots of matrix A after gaussian elimination

Free columns can be assigned any number as the solution doesn’t depend on these columns whatever the amount of component these columns try to add can be removed by the pivot columns as these free columns are linearly dependent on the pivot columns

free variables = no of columns - rank of the matrix

reduced row echelon form is basically as many zeros possible above the matrix as well and pivots should be 1

no of special solutions = no of free variables

give free variables any value and calculate the special solutions

Block matrix multiplication view here N is a null space matrix whose columns are the special solutions of Ax=0

F is the free variables columns with values as a matrix

What matrix can be multiplied to R to get a zero matrix the matrix has to be the null space matrix

and this gives us how to calculate the null space matrix as N = [-F I ] multiplied to R would give the zero matrix

RREF =[[I,F],[0,0]] here first I is rr matrix F is a r(n-r) matrix first 0 is a (m-r)r and the second 0 is a (m-r)(n-r). in RN=0 for N=[-F,I] -F is a r*(n-r) and I is (n-r)*(n-r) matrix