2.12 Tensor Vector Addition Operation

Tensor Vector Addition Operation is the binary operation defined on a vector space used for tensor construction that takes two vectors and produces a third vector, satisfying the algebraic laws required of every vector space, and forming, together with scalar multiplication, the pair of fundamental operations from which all linear algebraic structure is built. Vector addition is the first and most basic operation whose behavior tensor algebra must respect at every level, from individual vectors up through fully constructed tensors.

Formal Statement

Definition as a Binary Operation

Vector addition takes any two vectors from the space and produces another vector from the same space, satisfying the closure requirement built into the definition of a vector space.

Required Algebraic Behavior

As part of the vector space axioms, addition must be commutative, associative, admit an additive identity, and admit an additive inverse for every vector.

Behavior in Coordinates

Component Form of Addition

Once a basis is fixed, vector addition corresponds exactly to entrywise addition of the coordinate vectors of the two operands, so the coordinate representation of a sum is obtained by adding corresponding components.

Compatibility With Any Choice of Basis

This entrywise behavior in coordinates holds for every choice of basis, though the specific numeric values involved depend on the basis, since addition itself is defined at the level of the abstract vectors and only mirrored, not altered, by coordinate representation.

Structural Role Within the Vector Space

Closure as a Defining Feature

The closure of the addition operation, guaranteeing the sum of two vectors from the space remains in the space, is inseparable from the very definition of a vector space and underlies span closure arguments used throughout the theory.

Interaction With Scalar Multiplication

Vector addition interacts with scalar multiplication through distributive laws, ensuring that scaling a sum of vectors produces the same result as summing the individually scaled vectors, a compatibility that is essential to the overall linear structure of the space.

Role in Tensor Construction

Supplying the Additive Structure of Tensor Spaces

The addition operation defined on each factor vector space is what allows the tensor space built from those factors to itself be equipped with a well-defined addition operation, since tensor addition is built up from, and must remain consistent with, the addition operations of the contributing spaces.

Serving as an Input to Multilinear Constructions

Vector addition acts as one of the two operations, alongside scalar multiplication, whose behavior any multilinear map used in tensor construction must respect separately in each of its input slots.

Summary of Key Properties

Fundamental Binary Operation of the Vector Space

Tensor Vector Addition Operation establishes the core binary operation from which the additive structure of the entire vector space, and by extension of tensors built from it, is derived.

Consistent Behavior Across Abstraction Levels

The algebraic laws governing vector addition remain consistent whether reasoning abstractly about vectors, concretely about coordinate tuples, or at the higher level of tensors constructed from multiple vector spaces.