linearity

One of the axioms regarding lines and planes in particular: if a plane \[\mathbf{P}\] contains points \[A\] and \[B\], then it also contains the line \[AB\], expresses the intuitive notion of "flatness". Non-planar surfaces do not satisfy this property: for example of it you take two points on the surface of a sphere, the line joining those points does not lie on the surface; rather, it cuts through the interior of the sphere and exits the sphere. Inspired by this example, we might define the following property:

A subset \[S\] of \[\mathbb{R}^{n}\] is geometrically linear if for any two points \[a\] and \[b\] in \[S\], the entire line through those points is also contained in \[S\], i.e. if we take for any two points in \[S\], then every point of the form \[(1-t)a+tb,\,\forall t\in \mathbb{R}\] must also be in \[S\].

In simpler words, if \[S\] is geometrically linear, then \[S\] is a (usually) infinite set of points that when drawn out lies on a straight line/plane depending on the dimension. Also, the computation \[(1-t)a+tb\] is calculated coordinate-wise, e.g. in \[\mathbb{R}^{3}\] it is computed as \[((1-t)a_{1}+tb_{1},(1-t)a_{2}+tb_{2},(1-t)a_{3}+tb_{3})\].

The reason behind why the formula is \[(1-t)a+tb\] is actually pretty trivial. Imagine \[a\] and \[b\] as two parallel vectors on a piece of paper. Draw an arrow with it's tail at \[a\] and head at \[b\] and call that arrow as a vector \[c\]. Now, to go from \[a\] to \[b\], we would need to go along \[c\], which brings us to \[a+c\], or \[b\]. Similarly, if we were to say, go to the midpoint between \[a\] and \[b\], then we would go \[\frac{1}{2}\] of the way along \[c\], bringing us to \[a+\frac{1}{2}c\]. This is the sense in which \[\left\{ a+tc:t\in \left[ 0,1 \right] \right\}\] parametrices the line segment between \[a\] and \[b\]. Finally, \[a+tc=a+t(b-a)=a-ta+tb=(1-t)a+tb\], giving us our final equation. In our case, since we aren't just constrained between the segment between \[a\] and \[b\], our \[t\] is defined as all \[t\in \mathbb{R}\].

It is taught in high school algebra regarding polynomial functions of a single variable and it's relations to the graphs, e.g. \[f(x)=ax+b\] determines a line, \[g(x)=ax^{2}+bx+c\] determines a parabola, etc. More generally, one can consider the functions of more than one variable, e.g. \[f(x,y)=ax^{2}+bxy+cy^{2}+d\].

By analogy with the single-variable case, we can define algebraic linearity as: A multivariable polynomial function is algebraically linear if every term has a degree of 1. This is evidently stricter than the high school definition of "linear" in the sense that even a function such as \[f(x)=5x+3\] would be considered not algebraically linear as the term \[3\] has a degree of zero, despite \[f(x)\] being a straight line. Similarly, an expression like \[ax+by+cz\] is algebraically linear while \[ax^{2}+bxy+cz\] isn't.

It is a striking fact that the geometric and algebraic notion of linearity agree whenever both make sense. In familiar settings, say \[\mathbb{R}^{2}\] or \[\mathbb{R}^{3}\], taking any set of vectors and forming all their algebraically linear combination, i.e. their span, yields a set that is geometrically linear, e.g. a line or plane through the origin.

For contexts that defy a simple geometric picture the algebraic definition very much also applicable. For instance, consider three functions \[f(x)\], \[g(x)\] and \[h(x)\] defined on \[\mathbb{R}\]. You can form the set of "linear combinations", i.e. functions of the form \[af(x)+bg(x)+ch(x)\], which describes a set of functions that can be "built from" \[f\], \[g\] and \[h\] in an algebraic sense using only addition and scalar multiplication.