Question

# 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=

Forms of linear equations

Let $$AX = B$$ be a system of linear equations, where A is an $$m\times nm\times n$$ 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 $$AY = 0$$, and conversely any vector of the form $$X0+Y$$ is a solution.

2021-07-05

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 $$AX-AX0=b-b=0 \rightarrow A(X-X0)=0.$$

Conclude that $$X-X0=Y$$, therefore $$X=X0+Y$$
Now show that any vector of the form $$X0+Y$$ is a solution. $$A(X0+Y)=AX0+AY=b+0=b$$