## Polynomials of linear maps

Hi Professor Kim,

Could you help me with a (potentially trivial) question?

On page 28 of the notes in the proof of Lemma 3.4.80:

Why are you able to switch the order of f and g in the final equality when acting on w1? Is it due to w1 being in the kernel of f(T)?

Many thanks

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Perhaps the notation was a bit misleading. refers to the linear map obtained by multiplying the polynomials and and then plugging the map into the place of the variable $x$. So the equality follows from the fact that .

On the other hand, it is easy to check that this is the same as composing with or with . The reason you can do it in either order is because the only maps appearing are , and ` commutes with ‘ so to speak. That is, if we had two different maps and , we definitely wouldn’t have

in general.

To give a formal proof of the equality

first do it for , which is easy, then just work it out for the general form of :

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