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 Incoming links: Basis of the space of real skew-symmetric matrices

Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 6 / a / Solution 2026-10-25

Let

E_{ij}

denote a matrix unit, with its only nonzero entry equal to one in row

i

, column

j

. A basis for the vector space of real skew-symmetric matrices is

{E_{ij} - E_{ji} : 1 \leq i < j \leq n}, dim = \frac{n ( n - 1 )}{2} .

Each upper-triangular entry is free, its reflected lower-triangular entry is its negative, and every diagonal entry is zero. This is the basis of the space of real skew-symmetric matrices.

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