Problem-5

system of equations

Question

\(x_{1}\) and \(x_{2}\) are two distinct solutions of \(Ax=b\). Find at least one solution for each of the following systems:

  • \(Ax=0\)
  • \(Ax=2b\).

Hint

Add and subtract the two systems.

Solution

Adding the two systems: \[ \begin{equation*} \begin{aligned} Ax_{1} & =b\\ Ax_{2} & =b \end{aligned} \Longrightarrow \begin{aligned} Ax_{1} +Ax_{2} & =2b\Longrightarrow A( x_{1} +x_{2}) =2b \end{aligned} \end{equation*} \]

\(x_1 + x_2\) is a solution of the system \(Ax = 2b\).

Subtracting the two systems: \[ \begin{equation*} \begin{aligned} Ax_{1} & =b\\ Ax_{2} & =b \end{aligned} \Longrightarrow A( x_{1} -x_{2}) =0 \end{equation*} \]

\(x_1 - x_2\) is a non-trivial solution of the system \(Ax = 0\). It should be clear that \(0\) is a trivial solution to \(Ax = 0\).

Note that we have abused the notations a bit here. \(x_{1}\) and \(x_{2}\) are vectors and not components of the vector \(x\).