8.38 min read

Linear Functions

A homogeneous linear function on $\mathbb{R}^n$ has the form $f(\mathbf{x}) = \mathbf{c}^\top \mathbf{x} = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$ for some fixed vector $\mathbf{c}$ . Every term involves exactly one variable to the first power (degree 1), and there is no constant offset. The function passes through the origin: $f(\mathbf{0}) = 0$ .

An affine function adds a constant offset: $f(\mathbf{x}) = \mathbf{c}^\top \mathbf{x} + d$ . Affine functions are sometimes loosely called "linear," but technically they are only truly linear (in the linear algebra sense) when $d = 0$ . The graph of an affine function on $\mathbb{R}^2$ is a tilted plane.

In matrix notation, the vector $\mathbf{c}$ is called the coefficient vector, and $\mathbf{c}^\top \mathbf{x}$ is just a dot product. As we will see in Chapter 12, $\mathbf{c}$ will turn out to be the gradient of the linear function — the direction of steepest ascent.

Formal View

Definition 8.3 — Linear and Affine Functions

A function

f: \mathbb{R}^n \to \mathbb{R}

is homogeneous linear if

f(\mathbf{x}) = \mathbf{c}^\top \mathbf{x}

for some

\mathbf{c} \in \mathbb{R}^n

. It is affine if

f(\mathbf{x}) = \mathbf{c}^\top \mathbf{x} + d

for some

\mathbf{c} \in \mathbb{R}^n

and

d \in \mathbb{R}

. Both have polynomial degree 1 (assuming

\mathbf{c} \neq \mathbf{0}

Homogeneous linear functions satisfy $f(\alpha \mathbf{x}) = \alpha f(\mathbf{x})$ and $f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y})$ . Affine functions satisfy neither unless $d = 0$ .

Interactive Visualization

Matrix-Vector Multiplication

Why This Matters

Linear functions are the foundation of calculus: every smooth function looks linear when you zoom in close enough (this is the local linear approximation idea).

Linear regression predictions: $\hat{y} = \mathbf{w}^\top \mathbf{x}$ is an affine function of the weight vector $\mathbf{w}$ .
Signal processing: inner products $\mathbf{c}^\top \mathbf{x}$ compute projections and filter responses.
Economics: linear utility functions $U(x_1, \ldots, x_n) = p_1 x_1 + \cdots + p_n x_n$ measure total value at prices $\mathbf{p}$ .

Learning Resources

Linear functions and gradients

MIT OpenCourseWare

Gilbert Strang connects linear functions to dot products and gradients.

40 min

Dot products and duality

3Blue1Brown

Geometric intuition for dot products, the foundation of linear functions.

14 min

Quiz

Question 1

Which of the following is a homogeneous linear function on $\mathbb{R}^3$ ?

Question 2

Every homogeneous linear function satisfies $f(\mathbf{0}) = 0$ .

Common Mistakes

Calling $f(\mathbf{x}) = 3x_1 + 5$ a linear function in the algebraic sense — it is affine, not homogeneous linear, because of the $+5$ .
Forgetting that the coefficient vector $\mathbf{c}$ is fixed, not a variable — $\mathbf{c}^\top \mathbf{x}$ is linear in $\mathbf{x}$ , not in $\mathbf{c}$ .