7.8.2 Tensor Vector Component Basis Coefficient

The Tensor Vector Component Basis Coefficient expresses vector components in a basis, crucial for tensor algebra and coordinate transformations.

Tensor Vector Component Basis Coefficient is the specific role a vector's component plays as the multiplier attached to one basis vector in the linear combination that reconstructs the vector, emphasizing the component's function as a weighting factor rather than as an isolated number.

Definition and Scope

Coefficient in a Linear Combination

Given a basis (e_1, \dots, e_n) of a vector space, any vector (v) is written uniquely as

v = \sum_{i = 1}^{n} v^{i} e_{i}

and each number (v^i) is the basis coefficient attached to the corresponding basis vector (e_i), telling how much of that particular basis vector contributes to building (v).

Uniqueness of the Coefficients

For a fixed basis, the coefficients (v^1, \dots, v^n) associated with a given vector (v) are unique: no other choice of coefficients paired with the same basis reproduces the same vector, a direct consequence of the basis vectors being linearly independent and spanning the space.

Structural Properties

Coefficients Depend Entirely on the Basis Chosen

Because the coefficients measure how much of each basis vector is needed, changing the basis changes every coefficient, even though the vector (v) itself is unaffected. A vector aligned exactly with one basis vector in one basis may require several nonzero coefficients when expressed in a different, rotated basis.

v = 1 e_{1} versus v = \frac{1}{\sqrt{2}} e_{1}' + \frac{1}{\sqrt{2}} e_{2}'

Coefficients Under a Change of Basis

The precise rule governing how basis coefficients change when the basis itself changes follows the standard contravariant transformation law, using the inverse of the matrix relating the new basis vectors to the old:

v^{'} = {(A^{- 1})}_{k}^{i} v^{k}

so that a rescaling of the basis vectors by a factor is compensated by the coefficients scaling inversely, keeping the reconstructed vector fixed.

Coefficient Values Reflect Geometry Relative to the Basis

The specific values of the basis coefficients reflect how the vector sits relative to the chosen basis directions: a coefficient of zero indicates the vector has no component along that basis direction, while a large coefficient indicates the vector extends substantially along it, a reading that is only meaningful once the basis is specified.

Role Within Tensor Algebra

Foundation of the Reconstruction Role

The basis coefficient interpretation is precisely what underlies the reconstruction role of a vector's components: reconstructing (v) from its components is nothing more than forming the weighted sum in which each basis coefficient multiplies its corresponding basis vector.

Extension to Higher-Rank Tensors

The notion of a basis coefficient generalizes directly to higher-rank tensors, where each component becomes the coefficient attached to a combination of several basis vectors and covectors formed via the tensor product, so the vector case's basis coefficient is the simplest instance of a pattern that recurs, with more basis elements involved, at every higher rank.