 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2010/2/a/solution

For real scalars

α, β

A, C, B

, distributivity of matrix multiplication and linearity of the trace give

φ (α A + βC, B) = tr (α A B + βCB) = α φ (A, B) + βφ (C, B) .

Thus $φ$ is linear in its first argument.

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