Test the following systems of equations as in Exercise 2. \[ \begin{array}{rl} -x_1 + x_2 + 4x_3 = 0, & x_1 - x_3 = 0, \\ x_1 + 3x_2 + 8x_3 = 0, & x_2 + 3x_3 = 0, \\ \frac{1}{2}x_1 + x_2 + \frac{5}{2}x_3 = 0. & \end{array} \]
Solution
Let \[ A_1=-x_1+x_2+4x_3, \quad A_2=x_1+3x_2+8x_3, \quad A_3=\frac12x_1+x_2+\frac52x_3, \] and \[ B_1=x_1-x_3, \qquad B_2=x_2+3x_3. \]
1. Every equation of the first system lies in the span of the second.
One has \[ A_1=-B_1+B_2, \qquad A_2=B_1+3B_2, \qquad A_3=\frac12B_1+B_2. \]
2. Every equation of the second system lies in the span of the first.
The identities \[ B_1=-\frac34A_1+\frac14A_2, \qquad B_2=\frac14A_1+\frac14A_2 \] hold by comparing coefficients of \(x_1,x_2,x_3\).
3. Therefore the two systems are equivalent.
By 1 and 2 the two sets of equations span the same subspace of the dual space, so their common zero sets coincide.