Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.
Solution
Regard each equation \(a x_1+b x_2=0\) as the linear functional \((a,b)\in F^2\). For a system \(\mathcal S\), let \(W_{\mathcal S}\le F^2\) be the span of its coefficient vectors.
1. The solution set of \(\mathcal S\) is\[ W_{\mathcal S}^{\perp} =\{x\in F^2:w\cdot x=0\text{ for every }w\in W_{\mathcal S}\}. \]
A vector \(x\) solves the system exactly when every coefficient vector occurring in the system annihilates \(x\). This is equivalent to every linear combination of those coefficient vectors annihilating \(x\).
2. For every subspace \(W\le F^2\), one has\[ (W^{\perp})^{\perp}=W. \]
Certainly \(W\subseteq(W^{\perp})^{\perp}\). Also \[ \dim W^{\perp}=2-\dim W \] for the nondegenerate standard pairing on \(F^2\), so \[ \dim (W^{\perp})^{\perp}=2-(2-\dim W)=\dim W. \] The inclusion is therefore equality.
3. Two homogeneous systems in two unknowns with the same solution set areequivalent.
If systems \(\mathcal S\) and \(\mathcal T\) have the same solution set, then by 1 \[ W_{\mathcal S}^{\perp}=W_{\mathcal T}^{\perp}. \] Taking perpendiculars and applying 2 gives \[ W_{\mathcal S}=W_{\mathcal T}. \] Thus every equation of either system is a linear combination of equations of the other, which is precisely equivalence of the systems.