Let A X = B be a system of linear equations, where A is m × n matrix and X is n-vector, and B is m-vector. Assume that there is one solution X = X 0 . Show that every solution is of the form X 0 + Y, where Y is solution of the homogeneous system A Y = O, and conversely any vector of the form X 0 + Y is also a solution.To show the converse, I just have to check if X 0 + Y satisfies the equation which it does. How to show that the solution is of the form X 0 + Y?I am just guessing, Y will be in null space which is perpendicular to subspace space generated by row space of A. So X 0 is just projection of solution of the system in the subspace generated by row-space of A. Still I am not sure how to show this.

Question

Let   A  X  =  B be a system of linear equations, where   A is   m  ×  n matrix and   X is   n-vector, and   B is   m-vector. Assume that there is one solution   X  =      X    0  . Show that every solution is of the form       X    0    +  Y, where   Y is solution of the homogeneous system   A  Y  =  O, and conversely any vector of the form       X    0    +  Y is also a solution.To show the converse, I just have to check if       X    0    +  Y satisfies the equation which it does. How to show that the solution is of the form       X    0    +  Y?I am just guessing,   Y will be in null space which is perpendicular to subspace space generated by row space of   A. So       X    0   is just projection of solution of the system in the subspace generated by row-space of   A. Still I am not sure how to show this.

Larry Yates · Accepted Answer

Show that if   X is a solution, then   X  −      X    0   is a solution to the homogeneous system.

Let A X = B be a system of linear equations, where A is m ×<!-- ×

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