Difference Between Multiple Solutions of Non Homogeneous Linear Equations

Suppose we have a system of  
  non homogeneous linear equations  
\[\sum^m_{j=1} a_{ij}x_j=b_i , \: i= 1,...,n\]
Suppose we have two solutions  
The difference between these two solutions,  
  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  
\[\sum^m_{j=1} a_{ij}x_j=0 , \: i= 1,...,n\]

Add comment

Security code