6.12.4 Tensor One Zero Basis Behavior

Tensor One Zero Basis Behavior is the way in which the numerical components of a type one-zero tensor, an ordinary vector, respond to a change in the basis of the underlying vector space, this response consisting of the components being recombined according to the coefficients that relate the new basis vectors to the old, in a manner exactly opposite to how the basis vectors themselves are recombined. A single fixed vector, expressed in two different bases, yields two different arrays of components, and the basis behavior of a type one-zero tensor is precisely the rule connecting those two arrays.

How the Basis Determines the Components

Components as Coefficients in a Basis Expansion

A vector's components in a given basis are the coefficients required to write the vector as a linear combination of that basis's vectors, one coefficient for each basis direction. Changing which vectors are chosen as the basis changes what coefficients are needed to reconstruct the same fixed vector, and it is exactly this change in required coefficients that constitutes the basis behavior of the type one-zero tensor.

Opposite Recombination Compared to the Basis Vectors

If the new basis vectors are each written as linear combinations of the old basis vectors using a certain matrix of coefficients, the new components of a fixed vector are obtained from the old components using the inverse of that same matrix. This inverse relationship is what is meant by saying vector components transform contravariantly, that is, in a manner that runs opposite to, or "contrary to," the transformation of the basis vectors themselves, which is the origin of the term contravariant.

Consistency of Basis Behavior With Coordinate Transformation

The Two Descriptions Agree

The basis behavior described directly in terms of relating basis vectors and their coefficients is fully consistent with the coordinate-based transformation pattern that uses the direct Jacobian matrix, since a change of coordinates on the underlying space induces a corresponding change of basis on the tangent vectors at each point, with the Jacobian matrix playing the role of the coefficient matrix relating the two bases. The two viewpoints, one built from abstract basis vectors and one built from coordinate functions, describe the identical basis behavior from different but compatible starting points.

Fixed Vector, Varying Description

Throughout any change of basis, the vector itself, as an abstract geometric or algebraic object, does not move or change in any sense; only its numerical description changes. The basis behavior of a type one-zero tensor is therefore a statement purely about the relationship between two descriptions of one unchanging object, never a claim that the object undergoes any actual alteration.

Special Bases and Their Effect on Components

Orthonormal and Aligned Bases

Choosing a basis aligned with some natural structure present in a particular problem, such as an orthonormal basis aligned with the eigenvectors of an operator acting on the space, typically produces especially simple components for vectors that are themselves related to that structure, such as an eigenvector having only one nonzero component in such a basis. This simplicity is a direct consequence of basis behavior interacting favorably with the chosen basis, not an intrinsic property of the vector that would appear in every basis.

Basis Behavior and the Impossibility of a Universally Simple Description

Because basis behavior mixes components according to the coefficient matrix relating any two bases, no single basis can be expected to simplify every vector in a space simultaneously unless those vectors share some common structural alignment; a basis that simplifies one vector's components generally does nothing special for a second, unrelated vector, illustrating that basis behavior is a genuinely two-sided relationship between the vector and the basis chosen to describe it, not a property of the vector in isolation.

Consequences for Statements Involving Vectors

Basis-Independent Statements Require Care

Any statement asserting a relationship purely among the numerical components of one or more vectors, without reference to a full contraction against a compatible one-form or metric, is at risk of holding only in the particular basis in which it was written, since basis behavior generally alters those components under a change of basis. Statements intended to hold universally must instead be phrased in terms of the vectors themselves or in terms of invariant scalar quantities built from them.

Reconstructing the Vector From Any Basis

Given the components of a type one-zero tensor in one basis together with the coefficients relating that basis to another, the components in the second basis are fully and uniquely determined by the basis behavior rule, with no additional information required; this determinacy is what allows a vector defined once to be legitimately re-expressed in as many different bases as needed without any loss or ambiguity.