nLab
linear combination

Linear combinations

Idea

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).

Definition

Let RR be a rig, and let VV be a (left) RR-module. By ‘scalar’, we mean an element of RR; by ‘vector’, we mean an element of VV. Given any natural number nn and any nn-tuple (a 1,,a n)(a_1,\ldots,a_n) of scalars (so in short, given a finite list of scalars), we have an nn-ary operation on VV that maps (x 1,,x n)(x_1,\ldots,x_n) to

a 1x 1++a nx n. a_1 x_1 + \cdots + a_n x_n .

The result of this operation is the linear combination of the vectors x 1x_1 through x nx_n with respective coefficients a 1a_1 through a na_n.

Variations: If RR is a non-unital rig (or more generally if MM is a non-unital module), then a linear combination may also have a term x 0x_0 with no coefficient. If RR is non-associative (or more generally if MM is a non-associative module), then the term with x ix_i takes the form

a i,1(a i,2(a i,i mx i))). a_{i,1} (a_{i,2} \cdots (a_{i,i_m} x_i){\cdots})) .

If RR is non-commutative and VV is a right RR-module, the term with x ix_i takes the form x ia ix_i a_i. If VV is an RR-SS-bimodule, then the term with x ix_i takes the form a ix ib ia_i x_i b_i, with a i:Ra_i\colon R and b i:Sb_i\colon S. Of course, all of these variations may be combined.

Given a subset GG of (the underlying set of) VV, the set of all linear combinations of the vectors in GG is a submodule of VV, the RR-linear span of GG.

Examples

Every operation on the module VV is a linear combination:

  • The identity operation is the linear combination of arity 11 with coefficient 11.

  • Addition is the linear combination of arity 22 with coefficients (1,1)(1,1), and the zero element is the linear combination of arity 00 (with no coefficients).

  • Scalar multiplication by the scalar aa is the linear combination of arity 11 with coefficient aa.

  • If RR is a ring (so 1-1 is a scalar), then subtraction is the linear combination of arity 22 with coefficients (1,1)(1,-1), and the additive inverse is the linear combination of arity 11 with coefficient 1-1.

  • If RR is divisible (so 1/n1/n is a scalar for every positive integer nn), then the mean? of nn vectors is the linear combination of arity nn with every coefficient 1/n1/n.

Special cases

An affine linear combination is a linear combination whose coefficients sum to 11. These are the operations in an affine space.

If RR 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.

Revised on August 29, 2012 19:38:56 by Toby Bartels (173.190.140.44)