linear combination: meaning, definition, pronunciation and examples
LowTechnical, Academic
Quick answer
What does “linear combination” mean?
In mathematics, an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.
Audio
Pronunciation
Definition
Meaning and Definition
In mathematics, an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g., a₁x₁ + a₂x₂ + ... + aₙxₙ, where a are coefficients).
A concept central to linear algebra denoting any sum of scalar multiples of vectors, matrices, or functions. In general usage, can metaphorically describe any situation where separate, simpler elements are combined in a proportional, additive manner to produce a more complex outcome.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning, spelling, or usage. The term is identical and used identically in both technical registers.
Connotations
None.
Frequency
Identically low frequency in both varieties, appearing almost exclusively in technical and academic contexts.
Grammar
How to Use “linear combination” in a Sentence
[Noun Phrase] + is a linear combination of + [Noun Phrase]to form/express + [Noun Phrase] + as a linear combination of + [Noun Phrase]Vocabulary
Collocations
Examples
Examples of “linear combination” in a Sentence
verb
British English
- We need to linearly combine these vectors to span the space.
- The function can be linearly combined from the basis set.
American English
- We need to linearly combine these vectors to span the space.
- You can linearly combine the columns to get the desired result.
adverb
British English
- (The term is not used as an adverb; 'linearly' is used instead, e.g., 'linearly independent').
American English
- (The term is not used as an adverb; 'linearly' is used instead, e.g., 'expressed linearly').
adjective
British English
- The linear-combination approach is fundamental.
- He studied the linear combination properties.
American English
- The linear-combination method is key.
- She explained the linear combination concept.
Usage
Meaning in Context
Business
Virtually unused. A metaphorical extension might be 'Our new strategy is a linear combination of our past successful approaches.'
Academic
Primary context. Found in mathematics, physics, engineering, computer science, and economics texts.
Everyday
Extremely rare. Would be confusing if used in general conversation.
Technical
The standard context. Used precisely to describe relationships in vector spaces, solutions to equations, signal processing, etc.
Vocabulary
Synonyms of “linear combination”
Strong
Neutral
Weak
Vocabulary
Antonyms of “linear combination”
Watch out
Common Mistakes When Using “linear combination”
- Using it to mean any simple combination. Incorrect: 'My lunch was a linear combination of a sandwich and an apple.'
- Confusing it with 'linear equation' (though they are related concepts).
- Pronouncing 'linear' as /laɪˈnɪə/ instead of /ˈlɪnɪə/.
FAQ
Frequently Asked Questions
Yes, almost exclusively. Its core and almost all of its usage is in mathematics and related technical fields like physics and engineering. Metaphorical use in everyday language is very rare and likely to cause confusion.
A 'combination' is any general joining or mixing. A 'linear combination' is a very specific mathematical operation: each element is multiplied by a number (a coefficient) and then the results are added together.
Coefficients are the scalar numbers (which can be real or complex) by which you multiply each vector, function, or term in the set before adding them all together. They determine the 'weight' or contribution of each element.
Yes. A single term like 5*v is considered a linear combination (where 5 is the coefficient and v is the vector), as it fits the definition: a sum of scalar multiples, even if the sum has only one element.
In mathematics, an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.
Linear combination is usually technical, academic in register.
Linear combination: in British English it is pronounced /ˈlɪnɪə ˌkɒmbɪˈneɪʃən/, and in American English it is pronounced /ˈlɪniər ˌkɑːmbɪˈneɪʃən/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “(none)”
Learning
Memory Aids
Mnemonic
Think of making a smoothie: a 'linear combination' is like adding specific amounts (coefficients) of different fruits (vectors) to get your final blend.
Conceptual Metaphor
BUILDING BLOCKS: Fundamental elements (vectors) are scaled (coefficients) and stacked/additively assembled to construct any entity within a given space.
Practice
Quiz
In which field is the term 'linear combination' MOST commonly used?