5.5 Tensor Elementary Tensor Structure

Tensor Elementary Tensor Structure is the detailed study of elementary tensors — the decomposable elements v ⊗ w of V ⊗ W — as a distinguished subset of the tensor product space, covering their symbolic form, the factors that constitute them, the order in which those factors are combined, and the role elementary tensors play in expanding and factoring general elements of the space.

What Distinguishes an Elementary Tensor

A Single Product of One Vector from Each Factor Space

An elementary tensor is an element of V ⊗ W of the specific form v ⊗ w, built from exactly one vector v in V and exactly one vector w in W, in contrast to a general element of V ⊗ W, which is a finite sum of such terms and need not itself reduce to a single one. Elementary tensors are also called pure tensors or decomposable tensors, and these terms are used interchangeably to refer to the same restricted class of elements.

Every Basis Element Is Elementary, But Not Conversely

The induced basis vectors e_i ⊗ f_j are elementary tensors, since each is v ⊗ w for v = e_i and w = f_j; but the class of elementary tensors is far larger than the basis, containing v ⊗ w for every pair (v, w), not merely for pairs of basis vectors, and it is not itself a subspace, since sums of elementary tensors are generally not elementary.

The Sub-Areas Covered

Symbolic Form

Concerns the precise notational conventions for writing an elementary tensor, v ⊗ w, and how this notation is distinguished from the space-level notation V ⊗ W and from sums of several such terms.

Factor Membership

Concerns which vector spaces the two components v and w of an elementary tensor belong to, and what it means for a component to range over the full space V or W rather than being restricted to a basis or a subspace.

Factor Order

Concerns the significance of the order in which v and w appear in v ⊗ w, and the sense in which v ⊗ w and w ⊗ v, when both are meaningful, denote elements of different spaces V ⊗ W and W ⊗ V rather than the same element written two ways.

Product Form

Concerns the specific bilinear combination rule that produces v ⊗ w from v and w, distinguishing it from other ways two vectors might be combined, and its relationship to the canonical map fixed during the tensor product's construction.

Expansion Role

Concerns how elementary tensors serve as the building blocks from which every element of V ⊗ W is assembled by finite summation, and the role this plays in proofs that proceed by first establishing a fact for elementary tensors and then extending it by linearity.

Pure Element Factorization

Concerns the question, for a given elementary tensor, of recovering its two factors v and w, and the extent to which this recovery is unique.

Why Elementary Tensors Are Studied as Their Own Structure

The Bridge Between the Factor Spaces and the Tensor Product

Elementary tensors are the only elements of V ⊗ W directly produced by the canonical bilinear map ⊗, and every other element is reached only by summing several of them; studying their structure in detail — how they are formed, ordered, and factored — is what makes precise the otherwise informal idea that V ⊗ W is built "out of" V and W.