EECS20N: Signals and Systems

Linear Functions

A function f: Reals → Reals is linear if ∀x∈ Reals and w ∈ Reals,

f(wx) = wf(x)

and ∀x ∈ Reals and y ∈ Reals,

f(x + y) = f(x) + f(y).

More compactly, f is linear if ∀x, y ∈ Reals and w, u ∈ Reals,

f(wx + uy) = wf(x) + uf(y).

Such a linear function has the form: ∀ x ∈ Reals

f(x) = ax

for some constant a.

More interestingly, f: Reals × Reals → Reals is linear if it has the form: ∀ s, x ∈ Reals

f(s, x) = as + bx

It is said to form a linear combination of its arguments.