Let \(F\) be the field of complex numbers. Are the following two systems of linear equations equivalent? If so, express each equation in each system as a linear combination of the equations in the other system. \[ \begin{array}{rl} x_1 - x_2 = 0, & 3x_1 + x_2 = 0, \\ 2x_1 + x_2 = 0, & x_1 + x_2 = 0. \end{array} \]
Solution
Write \[ A_1=x_1-x_2,\qquad A_2=2x_1+x_2, \] and \[ B_1=3x_1+x_2,\qquad B_2=x_1+x_2. \]
1. Each equation of the first system is a linear combination of theequations of the second system.
Direct calculation gives \[ A_1=B_1-2B_2 \] and \[ A_2=\frac12B_1+\frac12B_2. \] Thus every solution of the second system satisfies the first system.
2. Each equation of the second system is a linear combination of theequations of the first system.
Again by direct calculation, \[ B_1=\frac13A_1+\frac43A_2 \] and \[ B_2=-\frac13A_1+\frac23A_2. \] Thus every solution of the first system satisfies the second system.
3. Hence the two systems are equivalent.
By 1 and 2 they have exactly the same solution set, and each equation in either system lies in the linear span of the equations in the other.