Bilinear Form (Polynomial Theory)/Examples

Examples of Bilinear Forms

Arbitrary Example $1$

The expression:

$6 x y$

is a bilinear form with respect to $x$ and $y$.

Arbitrary Example $2$

Let $\set {x_1, x_2, \ldots, x_n}$ and $\set {y_1, y_2, \ldots, y_n}$ be sets of variables.

The expression:

\(\ds \)

\(\)

\(\ds a_{11} x_1 y_1 + a_{12} x_1 y_2 + \cdots + a_{1n} x_1 y_n\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds a_{21} x_2 y_1 + a_{22} x_2 y_2 + \cdots + a_{2n} x_2 y_n\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \vdots\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds a_{n1} x_n y_1 + a_{n2} x_n y_2 + \cdots + a_{nn} x_n y_n\)

is a bilinear form with respect to $\set {x_1, x_2, \ldots, x_n}$ and $\set {y_1, y_2, \ldots, y_n}$.