2.3.2 Tensor Vector Space Addition Operation

Explore how tensor vector space addition operates within algebra, combining vectors through tensor structures to define linear operations in multilinear algebra.

Tensor Vector Space Addition Operation is the operation that equips the underlying set of a tensor space with the structure of an abelian group by defining how two tensors of the same type combine to produce a third tensor of that same type. Given two tensors built from a vector space and its dual, this operation is defined componentwise: the sum of two tensors is the tensor whose components, in any fixed basis, are the sums of the corresponding components of the two operands. It is the additive counterpart to scalar multiplication, and together the two operations make the collection of tensors of a fixed type into a vector space in its own right.

Formal Definition

Setting

Let $V$ be a finite-dimensional vector space over a field $F$ , and let $T_{s}^{r} (V)$ denote the space of tensors of type $(r, s)$ over $V$ , meaning multilinear maps

T : \underset{r factors}{V^{*} \times \dots \times V^{*}} \times \underset{s factors}{V \times \dots \times V} \to F

Definition of the Sum

For two tensors $S, T \in T_{s}^{r} (V)$ , the sum $S + T$ is the map defined pointwise on the same domain:

(S + T) (ω^{1}, \dots, ω^{r}, v_{1}, \dots, v_{s}) = S (ω^{1}, \dots, v_{s}) + T (ω^{1}, \dots, v_{s})

The result is again multilinear in every argument, since a sum of multilinear maps is itself multilinear, so the sum is a well-defined element of the same tensor space.

Componentwise Description

Component Formula

Once a basis $e_{1}, \dots, e_{n}$ of $V$ and its dual basis are fixed, a tensor $T$ of type $(r, s)$ is represented by its components

T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}

The sum of two tensors then has components equal to the sum of the corresponding components:

{(S + T)}_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}} = S_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}} + T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}

Basis Independence

This componentwise rule is consistent across changes of basis: because both operands transform by the same multilinear transformation law under a change of basis, their sum transforms by that law as well, so addition does not depend on which basis is used to express the components.

Special Cases

Vectors

When $(r, s) = (1, 0)$ , tensors of this type are precisely the elements of $V$ itself, and the tensor addition operation reduces to ordinary vector addition in $V$ .

Covectors

When $(r, s) = (0, 1)$ , tensors of this type are linear functionals on $V$ , and the operation reduces to addition of linear functionals in the dual space $V^{*}$ .

Matrices and Bilinear Forms

When $(r, s) = (0, 2)$ , tensors correspond to bilinear forms, and in a fixed basis the addition operation coincides with ordinary entrywise matrix addition.

Algebraic Properties

Closure

For fixed type $(r, s)$ , the sum of two tensors of that type is again a tensor of that same type, so $T_{s}^{r} (V)$ is closed under this operation.

Commutativity and Associativity

Because addition in the field $F$ is commutative and associative, and the sum of tensors is defined via pointwise addition of scalar outputs, tensor addition inherits both properties:

S + T = T + S

(R + S) + T = R + (S + T)

Identity Element

The zero tensor, whose value is $0$ on every input tuple, acts as the identity element:

T + 0 = T

Inverse Elements

Every tensor $T$ has an additive inverse $- T$ , defined by negating every component, satisfying:

T + (- T) = 0

These four properties establish that $T_{s}^{r} (V)$ forms an abelian group under addition.

Relation to Vector Space Structure

Compatibility with Scalar Multiplication

Tensor addition does not stand alone: combined with scalar multiplication of a tensor by an element of $F$ , it satisfies the distributive laws expected of a vector space, namely

α (S + T) = α S + α T

(α + β) T = α T + β T

for scalars $α, β \in F$ . This compatibility is what promotes the set of tensors of a fixed type from a mere abelian group to a genuine vector space over $F$ .

Dimension Consequence

Since addition and scalar multiplication act componentwise relative to a basis induced from a basis of $V$ , the resulting tensor space has dimension $n^{r + s}$ , where $n$ is the dimension of $V$ , and behaves as an ordinary finite-dimensional vector space with respect to this addition operation.

Restriction Requirement

Addition is only defined between tensors sharing the same type $(r, s)$ ; tensors of differing type occupy different tensor spaces and cannot be added directly through this operation, since their multilinear maps have domains that do not match.