Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 8F / iv

Table of contents
- Solution iv

Solution

Statement (iv) is false because

α β = 0

need not imply

β α = 0

. For example, over any field take

α = (1000), β = (0101) .

Both maps are idempotent and

α β = 0

, but

β α = (0100) \neq = 0.

Consequently

(α + β)^{2} = α + β + β α \neq = α + β .

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