Question-8

Let \(v\) be a solution of the systems \(A_{1}x=b\) and \(A_{2}x=b\). Construct two systems that have \(v\) as a solution.

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As before, we can add these two systems to begin with. Since \(v\) is a solution to both systems:

\[ \begin{aligned} A_{1}v & =b\\ A_{2}v & =b\\ \implies A_{1}v+A_{2}v & =2b\\ \implies(A_{1}+A_{2})v & =2b \end{aligned} \]

Thus, \(v\) is a solution to the system \((A_{1}+A_{2})x=2b\). Next, subtracting the two systems:

\[ \begin{aligned} A_{1}v & =b\\ A_{2}v & =b\\ \implies A_{1}v-A_{2}v & =0\\ \implies(A_{1}-A_{2})v & =0 \end{aligned} \]

Thus, \(v\) is a solution to the system \((A_{1}-A_{2})x=0\).