linear operator: meaning, definition, pronunciation and examples
C2Technical/Academic
Quick answer
What does “linear operator” mean?
A mathematical function that maps between vector spaces while preserving vector addition and scalar multiplication.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical function that maps between vector spaces while preserving vector addition and scalar multiplication.
In functional analysis and linear algebra, a linear operator is a linear transformation from a vector space to itself (or between two vector spaces), fundamental to quantum mechanics, differential equations, and signal processing.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or grammatical differences; spelling follows national conventions for surrounding text.
Connotations
Identical technical connotations in both varieties.
Frequency
Equally frequent in academic mathematics and physics contexts in both regions.
Grammar
How to Use “linear operator” in a Sentence
The linear operator [acts/operates] on [vector/function].[Property] of the linear operator [is/are] [described].We [consider/define] a linear operator [symbol].Vocabulary
Collocations
Examples
Examples of “linear operator” in a Sentence
verb
British English
- The transformation linearises the system.
- We need to linearise the model around the equilibrium point.
American English
- The transformation linearizes the system.
- We need to linearize the model around the equilibrium point.
adverb
British English
- The system behaves linearly within this range.
- The components are coupled linearly.
American English
- The system behaves linearly within this range.
- The components are coupled linearly.
adjective
British English
- The operator's linearity is its defining feature.
- Linear operator theory is a major branch of functional analysis.
American English
- The operator's linearity is its defining feature.
- Linear operator theory is a major branch of functional analysis.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core term in mathematics, physics, and engineering, especially in functional analysis, quantum mechanics, and systems theory.
Everyday
Not used in everyday conversation.
Technical
Precise term for a specific type of function between vector spaces.
Vocabulary
Synonyms of “linear operator”
Vocabulary
Antonyms of “linear operator”
Watch out
Common Mistakes When Using “linear operator”
- Using 'linear operator' to describe a simple proportional relationship in business or everyday life.
- Confusing it with a linear *equation* or linear *function* in basic algebra.
- Omitting the requirement of additivity and homogeneity when defining it.
FAQ
Frequently Asked Questions
Yes, when considered as acting on a finite-dimensional vector space (like ℝⁿ) by matrix multiplication, a matrix defines a linear operator.
The terms are often used interchangeably, but 'linear operator' frequently implies the domain and codomain are the same vector space, while 'linear transformation' can be between different spaces.
No, that is a contradiction in terms. A 'linear operator' is defined by its linearity. A 'nonlinear operator' does not obey the principles of additivity and homogeneity.
They are fundamental in quantum physics (observables are operators), engineering (system responses), computer graphics (transformations), and any field using differential equations (derivatives are operators).
A mathematical function that maps between vector spaces while preserving vector addition and scalar multiplication.
Linear operator is usually technical/academic in register.
Linear operator: in British English it is pronounced /ˈlɪn.i.ər ˈɒp.ər.eɪ.tər/, and in American English it is pronounced /ˈlɪn.i.ɚ ˈɑː.pə.reɪ.t̬ɚ/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “None. The term is purely technical and does not feature in idiomatic expressions.”
Learning
Memory Aids
Mnemonic
Think 'LINE stays clear': A Linear operator keeps the rules of vector addition and scalar multiplication intact, so lines (vector structures) remain clear and predictable.
Conceptual Metaphor
A MACHINE THAT PRESERVES STRUCTURE: It is conceptualised as a precise machine that takes in vectors, stretches, shrinks, rotates, or reflects them, but never bends or breaks the fundamental grid-like (linear) structure of the space.
Practice
Quiz
Which property is NOT necessarily true for all linear operators?