 # Let AX = B be a system of linear equations, where A is an m×nm×n matrix, X is an n-vector,and BB is an m-vector. Assume that there is 1 solution X=X0 Brittney Lord 2021-09-17 Answered
Let AX = B be a system of linear equations, where A is an m×nm×n matrix, X is an n-vector, and BB is an m-vector. Assume that there is one solution X=X0. Show that every solution is of the form X0+Y, where Y is a solution of the homogeneous system AY = O, and conversely any vector of the form X0+Y is a solution.

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First show that every solution is of the form X0+Y where Y is a solution of the homogeneous system AY=0. Let X be one solution. AX=b. Then AX-AX0=b-b=0 -> A(X-X0)=0. Conclude that X-X0=Y, therefore X=X0+Y
Now show that any vector of the form X0+Y is a solution. A(X0+Y)=AX0+AY=b+0=b