10.3.3 Tensor Basis Transformation Coefficient Relation

Understanding how tensor basis transformation coefficients relate to coordinate changes in multilinear algebra.

Tensor Basis Transformation Coefficient Relation is the precise identity connecting each individual entry of the change-of-basis matrix to the coordinates of one basis expressed in terms of the other, clarifying exactly what a single matrix entry (A^i_{\ j}) means as a number rather than as an abstract symbol.

The Defining Relation

One Coefficient, One Coordinate

Each entry (A^j_{\ i}) of the change-of-basis matrix is, by definition, the coefficient of the old basis vector (e_j) appearing in the expansion of the new basis vector (e'_i).

e_{i}' = \sum_{j = 1}^{n} A_{i}^{j} e_{j}

This relation is what gives every entry of (A) a concrete meaning: it is not an arbitrary number but specifically the amount of (e_j) contained in (e'_i), extracted the same way any vector's components are extracted relative to a basis.

Extracting a Coefficient via the Dual Basis

Because each coefficient is itself a component of the vector (e'_i) relative to the old basis, it can be extracted directly using the old dual basis, applying the general component-reading technique to this specific case.

A_{i}^{j} = e^{j} (e_{i}')

Reading the Index Positions on the Coefficient

Why the Upper Index Matches the Old Basis

The upper index (j) on (A^j_{\ i}) matches the index of the old basis vector (e_j) that the coefficient multiplies, which is why this index later pairs, under the summation convention, with the lower index of an old covariant component during transformation.

Why the Lower Index Matches the New Basis Vector

The lower index (i) on (A^j_{\ i}) identifies which specific new basis vector (e'_i) the coefficient belongs to, tying that index to the label of the new basis vector being expanded.

A_{i}^{j} upper j: pairs with old basis lower i: labels new basis vector

The Inverse Coefficient Relation

Coefficients of the Old Basis in the New

The entries of (A^{-1}) satisfy the mirror-image relation, giving the coefficients of the old basis vectors expanded in terms of the new ones.

e_{i} = \sum_{j = 1}^{n} {(A^{- 1})}_{i}^{j} e_{j}'

Consistency Between A and Its Inverse

The coefficient relations for (A) and (A^{-1}) are not independent; substituting one relation into the other and using the fact that a basis expansion of a vector in its own basis is unique confirms that (A A^{-1}) must equal the identity matrix, tying the two coefficient relations together as consistent descriptions of the same pair of bases.

\sum_{j = 1}^{n} A_{i}^{j} {(A^{- 1})}_{j}^{k} = δ_{i}^{k}

Practical Reading of a Numerical Entry

An Entry as a Direct Numerical Fact

Given explicit coordinates for a new basis vector, the coefficient relation allows each entry of (A) to be read off immediately as a plain number: if (e'1 = 3e_1 + 2e_2), then (A^1{\ 1} = 3) and (A^2_{\ 1} = 2) directly, with no further computation required beyond identifying the coefficients in the expansion.

Visual Illustration

Why This Coefficient Relation Is Foundational

Understanding each matrix entry as literally a coefficient in a basis expansion, rather than as an abstract algebraic symbol, is what makes the change-of-basis matrix constructible and checkable directly from concrete basis data. This relation is the bridge between the geometric picture of two bases related by a linear map and the purely numerical matrix used mechanically throughout the rest of the transformation process.