10.8.3 Tensor Vector Component Basis Relation

Understanding how tensor components relate to their basis vectors in algebraic structures.

Tensor Vector Component Basis Relation is the pairing between a vector's components and the basis vectors they multiply, expressed as a sum in which each component scales exactly one basis vector, together with the requirement that this pairing reconstruct the identical vector whether the old basis and old components are used or the new basis and new components produced by the vector component change rule are used. It captures the precise sense in which components and basis vectors are linked to one another, and it is the relation from which the entire transformation behavior of contravariant components is derived.

The Relation Itself

The Defining Sum

A vector is expressed as the sum of its components, each multiplying the corresponding basis vector, with the summation convention implying an addition over every basis vector in the set.

v = v^{i} e_{i}

This equation is the basis relation itself: it states that the vector is nothing more or less than this particular linear combination of basis vectors weighted by components.

Invariance of the Sum Under a Change of Basis

The basis relation must continue to hold after a change of basis, with the new components paired against the new basis vectors reproducing the identical vector that the old components paired against the old basis vectors already produced.

v = v^{i} e_{i} = v^{i}' e_{i}'

Deriving the Component Change Rule From the Relation

Substituting the Basis Change

Substituting the forward expression for the new basis vectors in terms of the old basis vectors into the invariant sum produces an equation relating the old components, the new components, and the change-of-basis matrix.

v^{i} e_{i} = v^{i}' A_{i}^{j} e_{j}

Matching Coefficients on Each Basis Vector

Because the basis vectors are linearly independent, the coefficient of each individual basis vector on both sides of the equation must match exactly, and comparing these coefficients yields the vector component change rule directly, with the inverse matrix appearing once the equation is solved for the new components in terms of the old ones.

v^{j} = A_{i}^{j} v^{i}'

Structural Significance of the Relation

Basis Independence Rooted in a Single Equation

The entire apparatus of transforming vector components under a change of basis is a consequence of maintaining this single defining relation between components and basis vectors, rather than an independently imposed rule layered on top of it.

Uniqueness of Components Given a Basis

For a fixed basis, the basis relation determines the components of any given vector uniquely, since the linear independence of the basis vectors guarantees that only one set of coefficients can reproduce a given vector as their weighted sum.

Extension to Covectors and Higher-Rank Tensors

An analogous relation, pairing covariant components with dual basis covectors, and pairing mixed tensor components with combinations of basis vectors and dual basis covectors, extends the same underlying idea to every other type of tensor object.

Schematic Representation

The diagram shows two scaled basis vectors added head to tail, with the resulting blue arrow representing the vector defined precisely as the sum described by the basis relation.