---
tag: 0GJWQZCPZM
kind: statement
role: proposition
name: Trace is linear
msc: [15A15]
uses: [07PTPSZPME]
sources:
  - url: https://en.wikipedia.org/wiki/Trace_(linear_algebra)
    locator: Basic properties
---

## Statement

Let $R$ be a commutative ring, let $A$ and $B$ be $n \times n$ matrices over $R$, and let $c \in R$.
Then $\tr(A + B) = \tr A + \tr B$ and $\tr(cA) = c \tr A$, where $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

Thus the trace is a linear map from $M_n(R)$ to $R$.
Both identities follow at once from the definition, because the diagonal of $A + B$ is the sum of the diagonals.
Over a field, the trace is a [[linear form|07YQ1C78G0]] on the vector space of $n \times n$ matrices.

## Explanations

### L1 human

The trace of a sum is the sum of the traces.
The trace of a multiple is the same multiple of the trace.
This holds because we only add diagonal entries, and addition and scaling act entry by entry.