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

$\sum _{i=1}^{n}r_{i}s_{i}=0\Rightarrow r_{1}=\ldots =r_{n}=0.\,$

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 linearly independent spanning set.

Linearly independent sets in a module form a motivating example of an independence system.

References

• Victor Bryant; Hazel Perfect (1980). Independence Theory in Combinatorics. Chapman and Hall. ISBN 0-412-22430-5. 
• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 129-130. ISBN 0-201-55540-9.