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3.3 Linearity 89
3.3 LINEARITY

The concept of linearity is the underlying theme of our subject. In elementary
mathematics the term “linear function” refers to straight lines, but in higher
mathematics linearity means something much more general. Recall that a func-
tion f is simply a rule for associating points in one set D —called the domain
of f —to points in another set R —the range of f. A linear function is a
particular type of function that is characterized by the following two properties.

Linear Functions
Suppose that D and R are sets that possess an addition operation as
well as a scalar multiplication operation—i.e., a multiplication between
scalars and set members. A function f that maps points in D to points
in R is said to be a linear function whenever f satisfies the conditions
that
f(x + y)= f(x)+ f(y) (3.3.1)
and
f(αx)= αf(x) (3.3.2)
for every x and y in D and for all scalars α. These two conditions
may be combined by saying that f is a linear function whenever

f(αx + y)= αf(x)+ f(y) (3.3.3)

for all scalars α and for all x, y ∈D.

2
One of the simplest linear functions is f(x)= αx, whose graph in is a
straight line through the origin. You should convince yourself that f is indeed
a linear function according to the above definition. However, f(x)= αx + β
does not qualify for the title “linear function”—it is a linear function that has
been translated by a constant β. Translations of linear functions are referred to
as affine functions. Virtually all information concerning affine functions can
be derived from an understanding of linear functions, and consequently we will
focus only on issues of linearity.
3
In , the surface described by a function of the form
f(x 1 ,x 2 )= α 1 x 1 + α 2 x 2

is a plane through the origin, and it is easy to verify that f is a linear function.
For β
=0, the graph of f(x 1 ,x 2 )= α 1 x 1 + α 2 x 2 + β is a plane not passing
through the origin, and f is no longer a linear function—it is an affine function.