PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-11

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Trace is a linear functional on $M_{n} (F)$

Statement

For every natural $n$ , the map $tr : M_{n} (F) \to F$ is linear:

$tr (A + B) = tr (A) + tr (B), tr (λ A) = λ tr (A) .$

Facts & Assumptions

Given: Matrices $A, B \in M_{n} (F)$ and a scalar $λ \in F$ .

[L1]

The trace is the finite sum of the diagonal entries (The trace $tr (A)$ as the sum of the diagonal entries).

Proof

technique · direct

1.1

Entrywise addition and the finite-sum law give $tr (A + B) = \sum_{i < n} (a_{i i} + b_{i i}) = \sum_{i < n} a_{i i} + \sum_{i < n} b_{i i} = tr (A) + tr (B)$ .

givenL1

2.1

Likewise, $tr (λ A) = \sum_{i < n} λ a_{i i} = λ \sum_{i < n} a_{i i} = λ tr (A)$ .

step 1.1L1

3.1

If $n = 0$ , every displayed sum is empty and has value $0$ , so trace is the zero linear functional on the zero matrix space.

step 2.1L1∎

Depends on

Used by

Characters add on direct sums, multiply on tensor products, and conjugate on duals Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Axler, Linear Algebra Done Right, 4th ed., §8D (standard reference, not scraped)

Trace is a linear functional on Mn(F)Mn​(F)