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Let Af.
Since Af and P divides Af for Af, we have Af.
Thus the Af are projections which correspond to some direct-sum decomposition of the space V.
We wish to show that the range of Af is exactly the subspace Af.
It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.
Conversely, suppose that **ya is in the null space of Af.
If Af, then Af is divisible by Af and so Af, i.e., Af.
But then it is immediate that Af, i.e., that **ya is in the range of Af.
This completes the proof of statement ( A ).

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