15.2 Linear functions in multiple variables

One of the most important functions in mathematics is the linear function. In one variable, it takes the form l(x) = ax + b, where a and b are arbitrary real numbers.

We’ve seen linear functions several times already. For instance, Theorem 77 gives that differentiation is equivalent to finding the best linear approximation.

Linear functions, that is, functions of the form

are as important in multiple variables as in one.

To build up a deep understanding, we’ll take a look at the simplest case: a line on the two-dimensional plane.

Figure 15.5: A line on the plane

Given its normal vector m = (m₁,m₂) and its arbitrary point v₀, the vector x is on the line if and only if m and x −v₀ is orthogonal, that is, if

⟨m, x − v₀⟩ = 0 (15.1)

holds. (15.1) is called the normal vector equation of the line.

By using the bilinearity of the inner product and writing out ⟨m,x⟩...