 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-i/4/b/i/solution

E_{ij}

1 \leq i, j \leq n

, form a basis of

M_{n} (R)

: each matrix has the unique expression

A = \sum_{i, j} a_{ij} E_{ij}

. There are

n^{2}

of them, so

dim M_{n} (R) = n^{2} .

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