/sci/ - Science & Math

so its clear that [math]X_2[/math] is just made up of linear combinations of [math]X_1[/math], but i dont see how this makes [math]H_2[/math] a solution. for it to be a solution we have that [math]H_2*x_1 \neq H_2*x_1[/math] for every [math]x \in X[/math], but doesnt that fact that there are elements of [math]x_2[/math] that are representable by at least 2 linear combinations of elements of [math]X_1[/math] mean this doesnt hold true? or am i just overthinking it and that fact that everything in [math]X_2[/math] is a linear combination of [math]X_1[/math] is proof enough