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In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

Now the range of Af is the null space Af of Af.

Let us put Af.

By Theorem 10, D is a diagonalizable operator which we shall call the diagonalizable part of T.

Let us look at the operator Af.

Now Af Af so Af.

The reader should be familiar enough with projections by now so that he sees that Af and in general that Af.

When Af for each i, we shall have Af, because the operator Af will then be 0 on the range of Af.