4.3 Change of basis

Any invertible transformation $A$ on $V$ can be used to change the basis $e_{j}$ of $V$ to a new basis $e_{j}^{'}$ by taking

e_{j} \to e_{j}^{'} = A e_{j}

The matrix for a linear transformation $L$ transforms under this change of basis as

L_{j k} = e_{j}^{*} (L e_{k}) \to (e_{j}^{'})^{*} (L e_{k}^{'}) = (A e_{j})^{*} (L A e_{k}) = (A^{T})^{- 1} (e_{j}^{*}) (L A e_{k}) = e_{j}^{*} (A^{- 1} L A e_{k}) = (A^{- 1} L A)_{j k}

In the second step we are using the fact that elements of the dual basis transform as the dual representation. This is what is needed to ensure the relation

(e_{j}^{'})^{*} (e_{k}^{'}) = δ_{j k}

The change of basis formula shows that if two matrices $L_{1}$ and $L_{2}$ are related by conjugation by a third matrix $A$

L_{2} = A^{- 1} L_{1} A

then they represent the same linear transformation, with respect to two different choices of basis. Recall that a finite dimensional representation is given by a set of matrices $π (g)$ , one for each group element. If two representations are related by

π_{2} (g) = A^{- 1} π_{1} (g) A

(for all $g$ , $A$ does not depend on $g$ ) , then we can think of them as being the same representation, with different choices of basis. In such a case the representations $π_{1}$ and $π_{2}$ are called “equivalent”, and we will often implicitly identify representations that are equivalent.

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正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 37、38

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

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4.3 Change of basis