Linear independence is a property of sets of vectors that tells whether or not any of the vectors can be expressed in terms of the other vectors (and any scalars).
A linear combination of elements in a set of vectors in some vector space is a finite sum of scalar multiples of vectors in . If is a positive integer, are nonzero elements of the underlying field, and are vectors in , then
is a linear combination.
Note that could occur, meaning that, for any set, the zero-vector can be a linear combination that is, by definition, trivial.
Linear combinations capture the concept of "reachable" vectors, vectors that could be reached by performing some finite number of vector space operations on the elements of . So the set of linear combinations of is the same as the set of "reachable" vectors of , and that set is a vector space itself, a subspace of . That set is known as the span of .
The question of whether or not a vector is a linear combination of other vectors factors in the discussion of the kernel and image.
Is a linear combination of and
Note that . So it is a linear combination of those two vectors.
Questions like these can be answered more generally with row reduction.
A set is known as linearly dependent if one of its elements is a linear combination of its other elements. In other words, there exists some vector such that .