1.5.3 Vector Tensor Representation

Vector Tensor Representation connects vectors and tensors, providing a framework to express multilinear relationships in mathematics and physics.

Vector Tensor Representation is the description of a tensor of type (1, 0), an object carrying a single upper index whose components transform contravariantly under a change of basis, mirroring the way the coefficients of a vector expressed in a basis change when that basis is replaced. It is the second member of the tensor type hierarchy after the scalar, and it establishes the reference pattern for what it means for an index to be contravariant, a pattern that recurs in every tensor carrying one or more upper indices.

Definition and Basic Structure

The Vector as a Type (1, 0) Tensor

A vector v belonging to a finite-dimensional vector space V is identified with the tensor space T^1_0(V), which is naturally identified with V itself. Relative to a basis e_1, ..., e_n of V, the vector is written as a linear combination of basis vectors, and the coefficients of that combination are its components.

v = v^{i} e_{i}

Why the Index Is Placed Above

The single index of a vector's components is written as a superscript to record that the vector belongs to the contravariant type, the type whose components transform using the change-of-basis matrix directly rather than its inverse. This placement is not decorative; it is the notational marker of the transformation law the component obeys.

The Contravariant Transformation Law

Deriving the Law from Basis Invariance

Since the vector v itself does not depend on which basis is used to describe it, the equality v = v^i e_i must hold in every basis. Writing the new basis vectors e′_j in terms of the old ones through a change-of-basis matrix A, and demanding that the sum reproduce the same vector v, forces the new components v′^j to be built from the old components using the inverse of A.

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

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

Statement of the Law

The resulting law states that vector components transform by contraction with the change-of-basis matrix acting on the single upper index, which is the defining property used to classify any single-index array of numbers as a genuine vector rather than an arbitrary list.

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

Geometric Interpretation

Direction and Magnitude Independent of Coordinates

A vector is often pictured as an arrow with a definite length and direction in space, and this picture is exactly what the component transformation law protects: the arrow itself does not move or change length when a new basis is chosen, only the numbers used to describe it relative to that basis change.

Vectors as Directional Derivatives

In the setting of smooth manifolds, vectors are frequently represented instead as directional derivative operators acting on functions, an equivalent formulation in which the components v^i appear as coefficients multiplying the partial derivative operators associated with a coordinate system, preserving the same contravariant transformation law.

Relation to Other Tensor Types

Contrast with the Covector

Where a vector's components transform with the matrix A, a covector's components transform with the inverse matrix A^{-1} acting on a lower index, making the two representations dual to one another: pairing a vector with a covector through contraction produces a scalar precisely because the two transformation factors cancel.

ω_{i} v^{i} = ω_{'} i v^{'} i

Vectors as Building Blocks of Higher Rank Tensors

Every tensor of type (p, q) is constructed by repeated tensor products of p copies of the vector representation and q copies of the covector representation, so the vector tensor representation is one of the two elementary pieces, alongside the covector, from which all component tensor representations are assembled.

Diagrammatic Summary

The diagram highlights the essential fact of vector tensor representation: an upper index whose components transform through direct multiplication by the change-of-basis matrix A, keeping the underlying vector v fixed while its numerical description adapts to whichever basis is in use.