6.19 Tensor Vector First Order Classification

Tensor Vector First Order Classification is the categorization of an ordinary vector as a tensor of type (1, 0), meaning it carries exactly one contravariant index and no covariant indices, placing it at the smallest nontrivial position in the tensor order hierarchy directly above the type (0,0) scalars. This classification formalizes the familiar notion of a vector, an element of a vector space V, as a special case of the general tensor concept, and it establishes the contravariant transformation law as the defining feature that distinguishes a vector from a covector, which occupies the parallel but distinct type (0,1) position.

Defining Features of the Vector Classification

Index Structure

A type (1,0) tensor is written with a single superscript, v^i, reflecting that p = 1 and q = 0 in the general notation. Coordinate-free, a vector v is simply an element of V itself, since the tensor product defining type (1,0) objects, one factor of V and zero factors of V*, reduces to V alone.

A Single Free Index and n Components

Since a type (1,0) tensor has exactly one free index ranging over the dimension n of V, it has n independent components, v^1 through v^n, matching exactly the ordinary notion of a vector's coordinates relative to a chosen basis.

The Contravariant Transformation Law

Opposite Behavior to the Basis Vectors

Under a change of basis with transition matrix A, where e'_i = A^k_i e_k, the components of a vector transform using the inverse matrix B = A^{-1}:

This inverse relationship is precisely why the index is termed "contravariant," meaning it varies opposite, or counter, to the way the basis vectors themselves vary; when the basis vectors are scaled up by A, the components of a fixed vector must scale down by B in exactly the compensating way needed to keep the geometric object v = v^i e_i unchanged.

Why Invariance of the Vector Itself Requires This Law

The transformation law is not an arbitrary convention but a direct consequence of requiring the abstract vector v to remain the same geometric object regardless of the basis used to describe it: substituting the new basis and new components into v = v'^i e'_i and expanding using e'_i = A^k_i e_k must reproduce the original expression v = v^k e_k, which forces the components to transform by B exactly as shown.

The Vector as a Linear Functional on Covectors

Pairing with the Dual Space

The type (1,0) classification also reveals a functional role for vectors: each vector v can be regarded as a linear map on the dual space V*, sending a covector φ to the scalar φ(v). This double-duality perspective, treating V as acting on V* just as V* acts on V, is a standard feature of finite-dimensional vector spaces and underlies the symmetric treatment of vectors and covectors within the general (p, q) tensor framework.

Diagram of the Vector Classification

Distinguishing Vectors from Related First-Order Tensors

Versus Covectors of Type Zero One

A covector, or type (0,1) tensor, has one lower index instead, transforming with the forward matrix A rather than the inverse B; vectors and covectors both have total order one and both have n components, but they belong to different underlying spaces, V and V* respectively, and their opposite transformation behaviors are what allows their pairing φ_i v^i to produce a coordinate-invariant scalar.

Building Blocks for Higher-Order Tensors

Every higher-order tensor, whether type (2,0), (1,1), (0,2), or any type (p,q) with larger p and q, is built from tensor products of type (1,0) vectors and type (0,1) covectors, making the vector classification, alongside the covector classification, the fundamental generating building block of the entire tensor algebra, with the scalar classification serving as the base case beneath both.

Practical Role in Applications

The vector classification underlies the representation of directed quantities such as displacement, velocity, and force in physical applications, and its contravariant transformation law is what guarantees that a physical displacement remains the same physical displacement regardless of the coordinate grid used to measure it, differing only in the numerical values of its components from one coordinate system to another.