EECS20N: Signals and Systems

Linear Functions

A function f: RealsReals is linear if ∀xReals and wReals,

f(wx) = wf(x)

and ∀xReals and yReals,

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

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

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

Such a linear function has the form: ∀ xReals

f(x) = ax

for some constant a.

More interestingly, f: Reals × RealsReals is linear if it has the form: ∀ s, xReals

f(s, x) = as + bx

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