Definition

1. Adjective:
   • Linear with respect to each of two variables or positions: A mathematical property describing a function or expression that is linear when considered separately in each of two arguments or variables. This means if you hold one variable constant, the expression behaves as a linear function of the other variable.

Usage

• The term "bilinear" is primarily used in mathematics, physics, and engineering to describe specific types of functions, forms, or operations. It is a technical adjective and is not commonly used in everyday language.
• It modifies nouns such as "form," "map," "interpolation," "operation," or "function."

Examples

• Adjective:
  • The dot product is a classic example of a bilinear form.
  • The algorithm uses bilinear interpolation to estimate pixel values.
  • A bilinear map from two vector spaces to a third satisfies linearity in each argument separately.

Advanced Usage

• "Bilinear form": A specific type of function that takes two vectors and produces a scalar, linear in each argument.
  • Symmetric bilinear forms are fundamental in the study of inner product spaces.
• "Bilinear transformation": Also known as a Möbius transformation in complex analysis, it is a rational function of a specific form.
  • Conformal mappings can often be represented by a bilinear transformation.

Variants and Related Words

• Multilinear (adj): A generalization of "bilinear"; linear with respect to each of several variables.
  • The determinant is a multilinear function of the column vectors.
• Linearity (n): The property of being linear.
  • The bilinearity of the operation simplifies many proofs. (Note: "Bilinearity" is the noun form).

Synonyms

• Separately linear: (A descriptive phrase) Linear in each argument independently.
• Doubly linear: (Informal, less common) Emphasizing the two linear aspects.

Notes on Different Meanings

• "Bilinear" has a single, precise technical meaning. It does not have different everyday meanings or idiomatic uses. It is not used to form common phrasal verbs or idioms.

Adjective

1. linear with respect to each of two variables or positions