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
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asked 2021-07-04

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.

Answers (1)

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. \(A(X0+Y)=AX0+AY=b+0=b\)

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