 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof/5/a/solution

The zero matrix has the required pattern, and matrix addition and scalar multiplication preserve every prescribed zero entry. This proves the vector subspace condition. A basis consists of the matrix units

{E_{ii} : 1 \leq i \leq n} \cup {E_{i, i + 1}, E_{i + 1, i} : 1 \leq i < n} .

They independently select every permitted entry of a tridiagonal matrix, so the dimension is $n + 2 (n - 1) = 3 n - 2$ for

n \geq 1

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