Math 2.0

statement proposition

Trace is linear

Statement

Let RR be a commutative ring, let AA and BB be n×nn \times n matrices over RR, and let c∈Rc \in R. Then tr⁡(A+B)=tr⁡A+tr⁡B\tr(A + B) = \tr A + \tr B and tr⁡(cA)=ctr⁡A\tr(cA) = c \tr A, where tr⁡\tr is the trace.

Remarks

Thus the trace is a linear map from Mn(R)M_n(R) to RR. Both identities follow at once from the definition, because the diagonal of A+BA + B is the sum of the diagonals. Over a field, the trace is a linear form on the vector space of n×nn \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.

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Matrix, definition, no evidence levelMatrixdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionTrace is linear, statement, no evidence levelTrace is linearstatement
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Sources

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Basic properties

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