\[n\]
non homogeneous linear equations \[\sum^m_{j=1} a_{ij}x_j=b_i , \: i= 1,...,n\]
.Suppose we have two solutions
\[(x_1,...,x_m)\]
and \[(x'_1,...,x'_m)\]
.The difference between these two solutions,
\[(x'_1-x_1,...,x'_m-x_m)\]
is a solution to the homogeneous system of equations \[\sum^m_{j=1} a_{ij}x_j=0 , \: i= 1,...,n\]
.Notice that these solutions satisfy
\[\sum^m_{j=1} a_{ij}x_j=b_i , \: \sum^m_{j=1} a_{ij}x'_j=b_i, \: i= 1,...,n\]
.Subtracting these gives the homogeneous system
\[\sum^m_{j=1} a_{ij}(x'_j-x_j)=0, \: i= 1,...,n\]
which is equivalent to the system \[\sum^m_{j=1} a_{ij}x_j=0 , \: i= 1,...,n\]
.