linear dependence: meaning, definition, pronunciation and examples
LowHighly Formal / Academic / Technical
Quick answer
What does “linear dependence” mean?
In linear algebra, a relationship where a set of vectors can be expressed as a linear combination of one another, meaning at least one vector in the set is redundant.
Audio
Pronunciation
Definition
Meaning and Definition
In linear algebra, a relationship where a set of vectors can be expressed as a linear combination of one another, meaning at least one vector in the set is redundant.
More broadly, a relationship or correlation between two or more variables, systems, or concepts where a change in one directly produces a proportional change in another. The concept implies a lack of independence and a specific, predictable connection.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage; the term is identical in both varieties. Spelling conventions follow local norms (e.g., 'analyse' vs. 'analyze' in surrounding text).
Connotations
Purely technical and formal with no regional connotative difference.
Frequency
Equally low-frequency and specialized in both regions, confined to STEM contexts.
Grammar
How to Use “linear dependence” in a Sentence
[Subject: vector set] + exhibits + linear dependence[Subject: analysis] + shows + the linear dependence + of [object: variables]There is + linear dependence + between X and YVocabulary
Collocations
Examples
Examples of “linear dependence” in a Sentence
verb
British English
- The vectors were found to linearly depend on each other.
- We need to check if these signals linearly depend.
American English
- The data points linearly depend on the primary variable.
- Do these equations linearly depend on one another?
adverb
British English
- The outcomes varied linearly dependently with the inputs.
American English
- The systems failed linearly dependently.
adjective
British English
- The vectors have a linearly dependent relationship.
- We identified a linearly dependent set of equations.
American English
- The system is linearly dependent.
- They are linearly dependent variables.
Usage
Meaning in Context
Business
Rare; might appear in advanced econometrics or quantitative finance reports to describe correlated market variables.
Academic
Core term in mathematics, physics, engineering, statistics, and computer science courses and papers.
Everyday
Virtually never used.
Technical
The primary domain. Used to describe properties of vectors, equations, signals, or datasets in mathematics, engineering, and data science.
Vocabulary
Synonyms of “linear dependence”
Strong
Neutral
Vocabulary
Antonyms of “linear dependence”
Watch out
Common Mistakes When Using “linear dependence”
- Using it as a fancy synonym for general 'dependence' in non-technical writing.
- Confusing 'linear dependence' with 'statistical correlation' – while related, the former is a stricter, deterministic mathematical property.
FAQ
Frequently Asked Questions
The opposite is 'linear independence,' where no vector in a set can be written as a linear combination of the others.
In its strictest sense, it is defined for vectors. However, the concept is extended metaphorically to functions, equations, data sets, and variables in other technical fields.
No. Linear dependence describes a mathematical relationship, not necessarily causation. Two variables can be linearly dependent (correlated) without one causing the other.
A common method is to put the vectors as columns in a matrix and calculate the determinant or row-reduce the matrix. If the determinant is zero or if you get a row of zeros, the vectors are linearly dependent.
In linear algebra, a relationship where a set of vectors can be expressed as a linear combination of one another, meaning at least one vector in the set is redundant.
Linear dependence is usually highly formal / academic / technical in register.
Linear dependence: in British English it is pronounced /ˌlɪnɪə dɪˈpɛndəns/, and in American English it is pronounced /ˈlɪniər dɪˈpɛndəns/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a line: points on a line depend on each other in a straight, predictable way. If vectors are 'linear dependents', they all lie on the same conceptual line or plane – one isn't going its own independent way.
Conceptual Metaphor
DEPENDENCE IS BEING ON THE SAME LINE / REDUNDANCY IS EXCESS BAGGAGE.
Practice
Quiz
In which field is the term 'linear dependence' MOST precisely and commonly defined?