Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 4 / b / i

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- Solution i

Solution

The matrix units

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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