DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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The basis-independent trace of an endomorphism of a finite-dimensional vector space

Definition

Let $V$ be a finite-dimensional vector space over $F$ and let $T : V \to V$ be linear. Choose any ordered basis $B$ of $V$ and define the trace of the endomorphism $T$ by

$tr (T) : = tr ([T]_{B}^{B}) .$

This is independent of the chosen basis: matrices of the same endomorphism in two ordered bases are similar, and similar matrices have equal trace. When $V$ is the zero space, the empty basis gives trace $0$ .

Depends on

Dependency tree · 2 levels

Sources

S. Axler, Linear Algebra Done Right, 4th ed., Definition 8.51