Question-7

Let \(x_{1}\) and \(x_{2}\) be solutions of the system \(Ax=b\). Consider the vectors \(x_{1}+x_{2}\) and \(x_{1}-x_{2}\). These two vectors are the solutions to which system?

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Since \(x_{1}\) and \(x_{2}\) are solutions to \(Ax=b\), \(Ax_{1}=Ax_{2}=b\). We can now perform two operations. The first is adding:

\[ \begin{aligned} Ax_{1}+Ax_{2} & =2b\\ A(x_{1}+x_{2}) & =2b \end{aligned} \]

Thus, \(x_{1}+x_{2}\) is a solution to the system \(Ax=2b\). Next, subtracting:

\[ \begin{aligned} Ax_{1}-Ax_{2} & =0\\ A(x_{1}-x_{2}) & =0 \end{aligned} \]

Thus, \(x_{1}-x_{2}\) is a solution to the system \(Ax=0\).