A function f: Reals ® Reals is linear if "xÎ Reals and w Î Reals,
f(wx) = wf(x)
and "x Î Reals and y Î Reals,
f(x + y) = f(x) + f(y).
More compactly, f is linear if "x, y Î Reals and w, u Î Reals,
f(wx + uy) = wf(x) + uf(y).
Such a linear function has the form: " x Î Reals
f(x) = ax
for some constant a.
More interestingly, f: Reals ´ Reals ® Reals is linear if it has the form: " s, x Î Reals
f(s, x) = as + bx
It is said to form a linear combination of its arguments.