Pronunciation: [ˌantɪlˈɪni͡ə] (IPA) 
The word "antilinear" is spelled as /æntiˈlɪn.iər/ in IPA phonetic transcription. The first syllable is pronounced as "an" with the short A sound, followed by "ti" with the short I sound. The second syllable is pronounced as "lin" with the long I sound and "iər" with the schwa sound. The word means "not linear" or "opposite to linear" and is commonly used in mathematics to describe functions that are not linear.
Antilinear is an adjective that describes a mathematical or physical operation that exhibits properties opposite to those of a linear operation. In other words, it refers to a function, operator, or transformation that does not adhere to the principles of linearity.
In mathematics, linearity refers to the property of a function or operator for which the superposition principle holds: the function behaves as a linear combination of its inputs. However, an antilinear operation violates this principle and instead exhibits the opposite behavior.
In the context of linear algebra, an antilinear transformation involves the reversal of scalar multiplication, that is, the transformation of a scalar multiplied by a vector or matrix will result in the vector or matrix multiplied by the complex conjugate of that scalar. This contrasts with linear transformations where scalar multiplication behaves normally.
Antilinearity can also be observed in quantum mechanics, particularly in the study of operators and wave functions. In this field, an antilinear operator is a transformation that reverses the order of multiplication, incorporating complex conjugation. Similarly, an antilinear wave function changes sign under interchange.
Overall, the concept of antilinear is used to describe mathematical operations, functions, transformations, or operators that exhibit behavior opposite to that of linearity, incorporating complex conjugation or reversals that deviate from the superposition principle in various mathematical and physical systems.
The word "antilinear" is a combination of two morphemes: "anti" and "linear".
The prefix "anti-" comes from the Greek "anti", meaning "against" or "opposite". It is commonly used to indicate opposition, contrast, or negation.
The word "linear" comes from the Latin word "linearis", which means "belonging to a line". It is derived from the Latin word "linea", meaning "line". In mathematics, "linear" refers to properties or entities that relate to or follow a straight line or have a direct proportionality.
When combined, the prefix "anti-" and the word "linear" create the term "antilinear", which implies something that is against or opposite to linearity. In the context of mathematics, "antilinear" typically refers to maps or transformations that preserve addition but reverse the scalar multiplication.
