2.5.1 Tensor Basis Vector Set

A tensor basis vector set provides a structured framework for representing tensors, enabling algebraic operations through linear combinations of basis elements.

Tensor Basis Vector Set is the finite, ordered set of linearly independent, spanning vectors of the underlying vector space that supplies the contravariant building blocks used to construct the induced basis of any tensor space built over that vector space. It is the more elementary of the two ingredient sets, the other being the dual basis covector set, from which every basis tensor of type $(r, s)$ is assembled by repeated tensor product.

Defining Properties

Setting

Let $V$ be a vector space of dimension $n$ over a field $F$ . A tensor basis vector set is an ordered set

(e_{1} e_{2} \dots e_{n})

of vectors in $V$ satisfying the two defining properties of a basis of $V$ : it spans $V$ , and it is linearly independent.

Spanning Property

Every vector $v \in V$ is expressible as a linear combination of the set:

v = v^{i} e_{i}

with implicit summation over $i$ from $1$ to $n$ , and coefficients $v^{i} \in F$ .

Linear Independence

No nontrivial linear combination of the set vanishes:

c^{i} e_{i} = 0 \Rightarrow c^{1} = \dots = c^{n} = 0

so that the representation of every vector as a linear combination of the set is unique.

Role as the Contravariant Ingredient

Filling the Vector Argument Slots

In constructing the induced basis of $T_{s}^{r} (V)$ , the tensor basis vector set supplies the factors occupying the contravariant, or upper-index, positions of each basis tensor product:

e_{i_{1}} \otimes \dots \otimes e_{i_{r}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{s}}

Each of the $r$ vector-type slots is filled independently by an element chosen from this same set, contributing a factor of $n$ possible choices per slot.

Complementary Dual Basis Covector Set

The tensor basis vector set determines a unique dual basis covector set $e^{1}, \dots, e^{n}$ of $V^{*}$ , defined by the pairing condition

e^{i} (e_{j}) = δ_{j}^{i}

which supplies the covariant, or lower-index, factors occupying the remaining $s$ slots of each basis tensor product.

Cardinality and Ordering

Fixed Cardinality

The tensor basis vector set always contains exactly $n$ elements, equal to the dimension of $V$ , since any two bases of a finite-dimensional vector space have the same cardinality.

Significance of the Ordering

Although a basis is set-theoretically unordered, the tensor basis vector set is conventionally treated as ordered, since the index labels $1, \dots, n$ attached to its elements are what allow components of a tensor and entries of a transformation matrix to be referenced unambiguously. Reordering the set permutes these labels and correspondingly permutes the components of every tensor expressed relative to it.

Behavior Under Change of Basis

Transformation to a New Set

A second tensor basis vector set ${\tilde{e}}_{1}, \dots, {\tilde{e}}_{n}$ is related to the first by an invertible matrix $A_{i}^{k}$ :

{\tilde{e}}_{k} = A_{k}^{i} e_{i}

Consistency Requirement

Any admissible replacement of the tensor basis vector set must itself satisfy the spanning and independence properties, which is guaranteed precisely when the matrix $A_{i}^{k}$ is invertible; a singular matrix would produce a set that fails to span $V$ or fails to be independent, and therefore does not qualify as a tensor basis vector set at all.

Propagation to the Induced Tensor Basis

Because every basis tensor of $T_{s}^{r} (V)$ is built from factors drawn from the tensor basis vector set and its dual, a change to this underlying set propagates through every contravariant factor of the induced basis, governed by the standard tensor transformation law with one factor of $A_{i}^{k}$ , or its inverse, per index.

Standard Examples

Coordinate Space

For $V = F^{n}$ , the standard tensor basis vector set consists of the columns of the identity matrix, each having a single entry of $1$ and all other entries $0$ .

Polynomial Spaces

For the space of polynomials of degree less than $n$ , a natural tensor basis vector set is the monomial set $1, x, \dots, x^{n - 1}$ , which likewise supplies contravariant factors for any tensor space built over that polynomial space.

Interaction with the Full Tensor Basis Construction

Necessity of Both Ingredient Sets

Neither the tensor basis vector set alone nor the dual basis covector set alone suffices to construct the induced basis of a mixed-type tensor space; both are required together whenever $r$ and $s$ are both positive, since the tensor basis vector set fills only the contravariant slots and cannot supply the covariant factors demanded by a nonzero $s$ .