nLab linear combination

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

$$a_1 x_1 + \cdots + a_n x_n .$$

The result of this operation is the linear combination of the vectors $x_1$ through $x_n$ with respective coefficients $a_1$ through $a_n$.

Variations: If $R$ is a non-unital rig (or more generally if $M$ is a non-unital module), then a linear combination may also have a term $x_0$ with no coefficient. If $R$ is non-associative (or more generally if $M$ is a non-associative module), then the term with $x_i$ takes the form

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

If $R$ is non-commutative and $V$ is a right $R$-module, the term with $x_i$ takes the form $x_i a_i$. If $V$ is an $R$-$S$-bimodule, then the term with $x_i$ takes the form $a_i x_i b_i$, with $a_i\colon R$ and $b_i\colon S$. Of course, all of these variations may be combined.

Given a subset $G$ of (the underlying set of) $V$, the set of all linear combinations of the vectors in $G$ is a submodule of $V$, the $R$-linear span of $G$.

More abstractly, by the adjunction between the underlying-set functor and the free functor, the subset inclusion

$$i_G\colon G \to {|V|}$$

corresponds to a homomorphism

$$\hat{i}_G\colon R[G] \to V .$$

Then the $R$-linear span of $G$ is the image of the homomorphism $\hat{i}_G$.

This abstract definition works more generally for any set function $f : G \to {|V|}$. The $R$-linear span of the image of $f$ in $V$ is the image of its corresponding homomorphism $\hat{f} \colon R[G] \to V$.

Examples

Every operation on the module $V$ is a linear combination:

• The identity operation is the linear combination of arity $1$ with coefficient $1$.

• Addition is the linear combination of arity $2$ with coefficients $(1,1)$, and the zero element is the linear combination of arity $0$ (with no coefficients).

• Scalar multiplication by the scalar $a$ is the linear combination of arity $1$ with coefficient $a$.

• If $R$ is a ring (so $-1$ is a scalar), then subtraction is the linear combination of arity $2$ with coefficients $(1,-1)$, and the additive inverse is the linear combination of arity $1$ with coefficient $-1$.

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

Special cases

An affine linear combination is a linear combination whose coefficients sum to $1$. These are the operations in an affine space.

If $R$ 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.

References

• William Lawvere, pp. 33 of: Introduction to Linear Categories and Applications, course lecture notes (1992) [pdf, pdf]