Let be a rig, and let be a (left) -module. By ‘scalar’, we mean an element of ; by ‘vector’, we mean an element of . Given any natural number and any -tuple of scalars (so in short, given a finite list of scalars), we have an -ary operation on that maps to
The result of this operation is the linear combination of the vectors through with respective coefficients through .
Variations: If is a non-unital rig (or more generally if is a non-unital module), then a linear combination may also have a term with no coefficient. If is non-associative (or more generally if is a non-associative module), then the term with takes the form
If is non-commutative and is a right -module, the term with takes the form . If is an --bimodule, then the term with takes the form , with and . Of course, all of these variations may be combined.
corresponds to a homomorphism
Then the -linear span of is the image of the homomorphism .
This abstract definition works more generally for any set function . The -linear span of the image of in is the image of its corresponding homomorphism .
Every operation on the module is a linear combination:
The identity operation is the linear combination of arity with coefficient .
Addition is the linear combination of arity with coefficients , and the zero element is the linear combination of arity (with no coefficients).
Scalar multiplication by the scalar is the linear combination of arity with coefficient .
If is a ring (so is a scalar), then subtraction is the linear combination of arity with coefficients , and the additive inverse is the linear combination of arity with coefficient .
If is ordered?, then a conical linear combination? is a linear combination whose coefficients are all positive, and a convex linear combination an affine conical linear combination. These are the operations in (respectively) a conical space and a convex space.