10 Tensor Change of Basis

Tensor Change of Basis is the study of how the components of a tensor must be recomputed when the basis of the underlying vector space is replaced by a different basis, together with the precise rules guaranteeing that the tensor being described remains exactly the same object throughout.

The Basic Setup

Two Bases Related by an Invertible Matrix

A change of basis begins with an original basis ({e_i}) and a new basis ({e'_i}) of the same vector space, related by an invertible matrix (A) whose entries express each new basis vector as a combination of the old ones.

Invertibility of (A) is essential: it guarantees that the new set is also a valid basis, spanning the same space with the same number of linearly independent vectors as the old one.

The Inverse Matrix for the Reverse Direction

Since (A) is invertible, its inverse (A^{-1}) expresses the old basis vectors in terms of the new ones, providing the reverse transformation needed whenever converting back from the new basis to the old.

Transformation of Components

Contravariant Components Transform with the Inverse

For a vector (v) to remain the same fixed object under this change of basis, its contravariant components must transform using (A^{-1}), the opposite of how the basis vectors themselves transform.

Covariant Components Transform with A Directly

Covector components, and the lower-index slots of a mixed tensor, transform directly with (A), matching the transformation of the basis vectors themselves.

General Mixed Tensors

A tensor with several indices transforms with one factor of (A) or (A^{-1}) for every index, applied according to whether that index is contravariant or covariant.

Why the Rules Take This Particular Form

Preserving the Tensor as an Invariant Object

The entire point of these transformation rules is to guarantee an invariance: the abstract tensor built by combining the new components with the new basis is identical to the one built from the old components and the old basis.

Substituting the transformation rules for both the basis and the components into this equation confirms it holds as an identity, which is the underlying reason the rules are defined exactly this way and not some other combination of (A) and (A^{-1}).

The Dual Basis Transforms Oppositely to the Primary Basis

Because the dual basis covectors (e^i) must continue to satisfy (e^i(e_j) = \delta^i_j) after the change, they are forced to transform with (A^{-1}), the same rule as contravariant components, which is why dual basis vectors and vector components share a transformation law while basis vectors and covector components share the other.

Visual Illustration

Practical Role

Tensor change of basis provides the computational bridge whenever a problem is more naturally expressed or solved in one basis but must ultimately be reported, compared, or combined in another. Every subsequent topic in tensor algebra, including invariants, contraction, and tensor operations, relies on the guarantee that quantities computed correctly in one basis will agree, once transformed, with the same quantities computed directly in another.