2.3.5 Tensor Vector Space Algebraic Laws

Tensor Vector Space Algebraic Laws define how tensors interact within vector spaces, establishing foundational rules for operations like multiplication and contraction.

Tensor Vector Space Algebraic Laws is the collection of identities that jointly govern how tensor addition and scalar action interact within the space of tensors of a fixed type, establishing that this space obeys the same algebraic laws as any abstract vector space over a field. These laws are not independent postulates layered on top of tensors; each one follows from the pointwise, componentwise nature of the two operations, yet stating them explicitly is what licenses treating tensors of a given type as ordinary vectors for all purposes of linear algebra.

Setting and Notation

The Ambient Space

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 $(r, s)$ . The laws below are asserted for arbitrary tensors $R, S, T \in T_{s}^{r} (V)$ and arbitrary scalars $α, β \in F$ .

Additive Laws

Commutativity of Addition

S + T = T + S

This holds because the sum is defined by adding scalar outputs of the field $F$ , and addition in $F$ is commutative.

Associativity of Addition

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

As with commutativity, this reduces to associativity of addition in $F$ , applied pointwise on every input tuple.

Additive Identity

There exists a zero tensor, the multilinear map sending every input tuple to $0$ , satisfying

T + 0 = T

for every $T$ .

Additive Inverses

Every tensor $T$ has an inverse $- T$ , obtained by negating every output value, such that

T + (- T) = 0

Together, these four laws make $T_{s}^{r} (V)$ an abelian group under addition.

Scalar Action Laws

Compatibility with Field Multiplication

α (β, T) = (α, β) T

Applying the scalar action twice in succession agrees with applying the scalar action once by the product of the two scalars in $F$ .

Multiplicative Identity

1 T = T

The multiplicative identity of $F$ acts as the identity transformation on every tensor.

Distributive Laws

Distributivity over Tensor Addition

α (S + T) = α S + α T

A single scalar distributes across a sum of two tensors.

Distributivity over Scalar Addition

(α + β) T = α T + β T

A single tensor distributes across a sum of two scalars. These two distributive laws together ensure that linear combinations of tensors, with coefficients drawn from $F$ , can be freely expanded and regrouped.

Componentwise Justification

Reduction to Field Arithmetic

Once a basis of $V$ is chosen, every tensor of type $(r, s)$ is represented by an indexed array of components in $F$ , and both addition and scalar action act entrywise on these components. Each of the algebraic laws above therefore reduces, coordinate by coordinate, to the corresponding field axiom applied to individual components, which is why no separate proof is needed beyond the arithmetic of $F$ itself.

Invariance Under Change of Basis

Since the component transformation law relating two bases is linear, and all of the laws above are linear identities in the tensors and scalars involved, verifying an algebraic law in one basis is sufficient to establish it in every basis; no law depends on the particular coordinate system used to express the tensors.

Consequence: Vector Space Status

Complete Axiom List

The additive laws, the scalar action laws, and the distributive laws together constitute the full list of vector space axioms. Their joint validity is precisely the statement that $T_{s}^{r} (V)$ , equipped with tensor addition and scalar action, is a vector space over $F$ of dimension $n^{r + s}$ , where $n$ is the dimension of $V$ .

Downstream Applicability

Because these laws hold, every construction available in general linear algebra, linear combinations, spanning sets, bases, linear independence, subspaces, and linear maps, applies without modification to tensors of a fixed type, forming the foundation on which further tensor algebra operations, such as the tensor product and contraction, are subsequently built.