Linear independence: Difference between revisions
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In [[algebra]], a '''linearly independent''' system of elements of a [[module (algebra)|module]] over a [[ring (mathematics)|ring]] or of a [[vector space]], is one for which the only [[linear combination]] equal to zero is that for which all the coefficients are zero (the "trivial" combination). | In [[algebra]], a '''linearly independent''' system of elements of a [[module (algebra)|module]] over a [[ring (mathematics)|ring]] or of a [[vector space]], is one for which the only [[linear combination]] equal to zero is that for which all the coefficients are zero (the "trivial" combination). | ||
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We have used the word "system" rather than "set" to take account of the fact that, if ''x'' is non-zero, the [[singleton]] set {''x''} is linearly independent, as is the set {''x'',''x''}, since this is just the singleton set {''x''} again, but the finite [[sequence]] (''x'',''x'') of length two is linearly dependent, since it satisfies the non-trivial relation ''x''<sub>1</sub> - ''x''<sub>2</sub> = 0. | We have used the word "system" rather than "set" to take account of the fact that, if ''x'' is non-zero, the [[singleton]] set {''x''} is linearly independent, as is the set {''x'',''x''}, since this is just the singleton set {''x''} again, but the finite [[sequence]] (''x'',''x'') of length two is linearly dependent, since it satisfies the non-trivial relation ''x''<sub>1</sub> - ''x''<sub>2</sub> = 0. | ||
A [[Basis (linear algebra)|basis]] is a linearly independent [[spanning set]]. | A [[Basis (linear algebra)|basis]] is a maximal linearly independent set: equivalently, a linearly independent [[spanning set]]. | ||
Linearly independent sets in a module form a motivating example of an [[independence | Linearly independent sets in a module form a motivating example of an [[independence structure]]. | ||
==References== | ==References== | ||
* {{cite book | author=Victor Bryant | coauthors=Hazel Perfect | title=Independence Theory in Combinatorics | publisher=Chapman and Hall | year=1980 | isbn=0-412-22430-5 }} | * {{cite book | author=Victor Bryant | coauthors=Hazel Perfect | title=Independence Theory in Combinatorics | publisher=Chapman and Hall | year=1980 | isbn=0-412-22430-5 }} | ||
* {{cite book | author=A.G. Howson | title=A handbook of terms used in algebra and analysis | publisher=[[Cambridge University Press]] | year=1972 | isbn=0-521-09695-2 | pages=40}} | * {{cite book | author=A.G. Howson | title=A handbook of terms used in algebra and analysis | publisher=[[Cambridge University Press]] | year=1972 | isbn=0-521-09695-2 | pages=40}} | ||
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=129-130 }} | * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=129-130 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:01, 12 September 2024
In algebra, a linearly independent system of elements of a module over a ring or of a vector space, is one for which the only linear combination equal to zero is that for which all the coefficients are zero (the "trivial" combination).
Formally, S is a linearly independent system if
A linearly dependent system is one which is not linearly independent.
A single non-zero element forms a linearly independent system and any subset of a linearly independent system is again linearly independent. A system is linearly independent if and only if all its finite subsets are linearly independent.
Any system containing the zero element is linearly dependent and any system containing a linearly dependent system is again linearly dependent.
We have used the word "system" rather than "set" to take account of the fact that, if x is non-zero, the singleton set {x} is linearly independent, as is the set {x,x}, since this is just the singleton set {x} again, but the finite sequence (x,x) of length two is linearly dependent, since it satisfies the non-trivial relation x1 - x2 = 0.
A basis is a maximal linearly independent set: equivalently, a linearly independent spanning set.
Linearly independent sets in a module form a motivating example of an independence structure.
References
- Victor Bryant; Hazel Perfect (1980). Independence Theory in Combinatorics. Chapman and Hall. ISBN 0-412-22430-5.
- A.G. Howson (1972). A handbook of terms used in algebra and analysis. Cambridge University Press, 40. ISBN 0-521-09695-2.
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 129-130. ISBN 0-201-55540-9.