Trace of a linear map

Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism. The trace of $T$ is $$\tr T = \tr [T]_B,$$ the trace of the matrix of $T$ in a basis $B$ of $V$.

Remarks

The value does not depend on $B$: by the change of basis formula, another basis gives $P^{-1} [T]_B P$, and $\tr(P^{-1} A P) = \tr A$. The trace is a linear map $\End_F(V) \to F$, and $\tr(ST) = \tr(TS)$. The trace of the identity map of $V$ is $\dim V \cdot 1$. A projection on a real or complex space has trace equal to its rank.

Explanations

L1 human

We define the trace of a linear map by its matrix in some basis. Another basis gives a different matrix, but the trace stays the same. So the trace belongs to the map, not to the basis.