When I am substituting variables of a polynomial, does that have any special meaning? Are there things I can't do or need to know about - before doing such a thing, to, say, solve an equation, or for anything else?
More specifically, when I read about Legrange's Reduction Method of Quadratic Forms, when regarding variable substitution, it usually says things like "operating a non-singular transformation on the variables".
Do they mean that if you look at the variables as polynomials such that: the variable which we place instead the substituted term, is the image of an invertible linear operator in $Hom(F _n [x],F _n [x])$ operating on the substituted terms?
If so, why is it that way? What exactly does that mean that it has to be invertible?
Let's see what happens if you do a transformation that is singular. Start with the form $$Q(x,y) = x^2 + xy + y^2$$
and make the singular transformation $$x = x_1+ y_1\\y = x_1 + y_1$$ The result is $$Q_1(x_1, y_1) = 3(x_1+y_1)^2$$
This may seem like an equivalent form, but it isn't. You can determine $Q_1$ from $Q$ by the transformation, but no transformation on $x_1, y_1$ will ever give you $Q$ back. $Q$ is not constant in any direction, but $Q_1$ is constant in the direction $(1,-1)$
That is why they talk about non-singular transformations. If the transformation is invertible, then you can recover the original form by another tranformation - the inverse of the original. But if the transformation is singular, then making the transformation loses information. The new form is more simple than the original, which cannot be recovered.
If this loss of complexity is not an issue for your application, then sure, you can use singular transformations. But generally it is undesirable, so they talk about non-singular transformations.