2.10.2 Tensor Coordinate Vector Basis Dependence

Tensor Coordinate Vector Basis Dependence is the property that the specific numeric entries of a coordinate vector depend entirely on which basis is used to describe the underlying vector space, so that the same abstract vector can be represented by different coordinate tuples under different bases even though the vector itself does not change. Basis dependence is a central caution in coordinate-based tensor work, since it means coordinate numbers alone are not meaningful without also specifying the basis they are relative to.

Formal Statement

Two Bases Yield Two Different Tuples

If a vector space carries two distinct bases, the coordinate vector of a fixed vector relative to the first basis is generally different from its coordinate vector relative to the second basis.

Relation via a Change of Basis Matrix

The two coordinate tuples for the same vector are related by an invertible change of basis matrix, which converts coordinates expressed in one basis into coordinates expressed in the other.

What Remains Invariant

The Vector Itself Is Unchanged

Although the coordinate tuple changes with the basis, the vector that the tuple represents is the same abstract object in the vector space regardless of which basis is chosen to describe it.

Dimension and Coordinate Length Stay Fixed

While individual component values change under a basis change, the length of the coordinate tuple, which equals the dimension of the space, remains the same across all bases, since dimension is a basis-independent invariant.

Practical Implications

Necessity of Specifying the Basis

Any statement about the coordinate vector of a given vector is incomplete unless it also specifies the basis relative to which that coordinate vector is expressed, since the numbers alone carry no meaning without this context.

Consistency Requirements in Computation

Computations that combine coordinate vectors, such as addition or the evaluation of a linear map, are only valid when all coordinate vectors involved are expressed relative to the same basis, making basis dependence a key source of errors when mismatched conventions are used.

Role in Tensor Construction

Basis Dependence of Tensor Components

Because tensors are built from coordinate vectors of their factor spaces, the components of a tensor inherit the same basis dependence, meaning a change of basis in any contributing vector space alters the numeric components of the tensor even though the tensor itself remains unchanged.

Motivation for Basis Change Formulas

Basis dependence is the underlying reason that explicit basis change formulas are needed throughout tensor algebra, since practitioners frequently need to translate coordinate or component data between different bases while preserving the represented object.

Summary of Key Properties

Coordinates as Basis-Relative Descriptions

Tensor Coordinate Vector Basis Dependence establishes that coordinate vectors are descriptions relative to a chosen basis rather than absolute properties of a vector, a distinction that must be tracked carefully throughout tensor work.

Governed by Invertible Linear Transformations

The way coordinate vectors change under a change of basis is governed by an invertible linear transformation, ensuring that basis dependence is systematic and reversible rather than arbitrary.