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Linear combinations are the most general operations in the operads for modules over a rig (including modules over a ring and vector spaces over a field).
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.
Given a subset of (the underlying set of) , the set of all linear combinations of the vectors in is a submodule of , the -linear span of .
More abstractly, by the adjunction between the underlying-set functor and the free functor, the subset inclusion
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 divisible (so is a scalar for every positive integer ), then the mean of vectors is the linear combination of arity with every coefficient .
An affine linear combination is a linear combination whose coefficients sum to . These are the operations in an affine space.
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.
Last revised on August 25, 2023 at 20:11:42. See the history of this page for a list of all contributions to it.