 Solution

ID: 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

Let

E_{ij}

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

i

, column

j

{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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