# Linear independence

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 *x*_{1} - *x*_{2} = 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.