Operad Theory
More abstractly, in the language of operad theory, one can consider vector spaces to be algebras over the operad (the infinite direct sum, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector for instance corresponds to the linear combination . Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.
From this point of view, we can think of linear combinations as the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that all possible algebraic operations in a vector space are linear combinations.
The basic operations of addition and scalar multiplication, together with the existence of an additive identity and additive inverses, cannot be combined in any more complicated way than the generic linear combination: the basic operations are a generating set for the operad of all linear combinations.
Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces.
Read more about this topic: Linear Combination
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