6.19.3 Tensor Vector Basis Component Form

Tensor Vector Basis Component Form is the expression of a vector as a linear combination of basis vectors weighted by its components, v = v^i e_i, providing the concrete, computable bridge between the abstract vector v, an element of V considered independently of any coordinate system, and the array of numbers v^i that represent it once a specific basis {e_i} has been chosen. This form is the practical starting point for essentially every calculation in linear and tensor algebra, since it is what allows abstract vectors to be added, scaled, and combined using ordinary arithmetic performed on their components.

Constructing the Basis Component Form

Existence and Uniqueness of the Expansion

Given a basis {e_1, ..., e_n} of an n-dimensional vector space V, every vector v in V can be written uniquely as:

compressed using the summation convention into v = v^i e_i. Existence follows from the defining property that a basis spans V, and uniqueness follows from the defining property that a basis is linearly independent, so no two distinct sets of coefficients can represent the same vector relative to the same basis.

Extracting Components Using the Dual Basis

Given the dual basis {e^j} satisfying e^j(e_i) = δ^j_i, the components of v can be recovered directly from the abstract vector by evaluation, v^j = e^j(v), which follows immediately from applying e^j to both sides of the basis component form and using the orthogonality relation between the basis and its dual.

Behavior of the Basis Component Form Under Vector Space Operations

Addition and Scalar Multiplication

The basis component form respects the vector space operations exactly: if v = v^i e_i and w = w^i e_i, then their sum satisfies v + w = (v^i + w^i) e_i, and for any scalar c, cv = (cv^i) e_i. These identities are what justify performing vector addition and scalar multiplication componentwise, treating the array of components as a stand-in for the vector itself once the basis is fixed.

The Zero Vector in Basis Component Form

The zero vector corresponds to the basis component form with every component equal to zero, 0 = 0 · e_i, and because of the uniqueness of the expansion, this is the only way to express the zero vector as a linear combination of a linearly independent basis, confirming that a vector is zero if and only if all of its components relative to any basis are zero.

Changing the Basis Component Form Under a Change of Basis

Recomputing Components in a New Basis

If the basis changes from {e_i} to {e'_i} via e'_i = A^k_i e_k, the same vector v acquires new components v'^i in the new basis component form v = v'^i e'_i, related to the old components by the contravariant transformation law v'^i = B^i_k v^k, where B = A^{-1}; this relationship is derived precisely by requiring the two basis component expansions, old and new, to describe the same underlying vector v.

Consistency Check via Substitution

Substituting the new basis vectors and new components into the new expansion and simplifying using BA = I reproduces the original expansion v = v^k e_k exactly, confirming that the basis component form, despite using different numbers in different bases, always describes one and the same invariant vector.

Diagram of the Basis Component Form

Extending the Basis Component Form to Other Tensors

Analogous Forms for Covectors and Higher Tensors

The basis component form generalizes directly to covectors, φ = φ_i e^i, and to tensors of every type (p, q), which expand as T = T^{...}_{...} e_{...} ⊗ e^{...}, using the appropriate basis products; the vector case discussed here is the simplest instance of this general pattern, and understanding its derivation, existence, uniqueness, and transformation behavior, provides the template for the analogous statements made about every higher-order tensor.

Practical Necessity for Computation

Without the basis component form, a vector remains an abstract element of V with no direct numerical handle; every practical computation, from solving a linear system to evaluating a physical quantity, proceeds by first fixing a basis, expressing the relevant vectors in basis component form, performing arithmetic on the resulting numbers, and, if needed, reinterpreting the numerical result back as an abstract vector using the same basis.