What is linear form?

Example 20.18 (An evaluation form and its hyperplane)

On $\R_2[X]$, the evaluation $\varphi(P) = P(2)$ is a linear form, nonzero ($\varphi(1) = 1$). Its kernel is the hyperplane of polynomials vanishing at $2$, i.e. (factor theorem, Theorem 8.7) the multiples of $X - 2$ within $\R_2[X]$:

In coordinates on $(1, X, X^2)$, $\varphi(a + bX + cX^2) = a + 2b + 4c$: every linear form on a finite-dimensional space is, once a basis is fixed, a fixed linear expression in the coordinates — forms are “row vectors”, as Chapter 21 will make literal, and the coefficient row here, $(1, 2, 4)$, is a Vandermonde row: evaluation forms are how the interpolation theory of Chapter 22’s weekend problem enters linear algebra.

Example 20.20

In $K^n$, a hyperplane is a solution set $\{a_1 x_1 + \dots + a_n x_n = 0\}$ with not all $a_i$ zero — the familiar equation of a plane through the origin in $\R^3$. In function spaces, evaluation forms $P \mapsto P(1)$ or $f \mapsto \int_0^1 f$ define hyperplanes of $\R_n[X]$, of $C(\intcc{0}{1})$ (cf. Exercise 19.6).

Example 20.21 (One hyperplane, worked three ways)

Take $\varphi(x, y, z) = x - 2y + 3z$ on $\R^3$ and $H = \ker\varphi$. Basis: solve $x = 2y - 3z$:

two free vectors: $\dim H = 2$, a hyperplane, as Theorem 20.19 predicts from $\varphi \neq 0$. Supplementary line: any vector outside $H$ spans one, e.g. $a = (1, 0, 0)$ ($\varphi(a) = 1 \neq 0$); the decomposition of an arbitrary $v$ is explicit:

since $\varphi\bigl(v - \varphi(v)a\bigr) = \varphi(v) - \varphi(v)\varphi(a) = 0$. Proportionality: if $\psi(x,y,z) = -2x + 4y - 6z$, then $\psi = -2\varphi$ and both have kernel $H$; conversely, any form vanishing on $H$ is a multiple of $\varphi$ (Exercise 20.8) — the equation of a hyperplane is unique up to scale, a fact used constantly for planes in geometry.