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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.50 Linear Operators andTheir Properties

1 Some Examples of Linear Operators

A LINEAR MAP [I.3 §4.2] between two VECTOR SPACES [I.3 §2.3] V and W is a function T: V → W that satisfies the condition T1υ1 + λ2υ2) = λ1Tυ1 + λ2Tυ2. Two phrases that are used almost interchangeably with “linear map” are “linear transformation” and “linear operator.” The former is often used when one wishes to draw attention to the effect of a linear map on some other object; for example, one might well choose to use the word “transformation” to describe geometrical operations such as reflections or rotations. As for “operator,” it tends to be the word of choice when the linear map is between infinite-dimensional spaces, especially when it is just one of an ensemble ...

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