/sci/ - Science & Math

>>11469208
Try proving that [math]f(z)g(z)=h(z)[/math] implies [math]f(A)g(A)=h(A)[/math] and using that to factor the polynomial and induct on dimension.
>>11469373
Basically, consider the existence theorem for bases of vector spaces.
It doesn't produce "this specific, canonical" base, it produces "some" base, which can be thought of as "randomly chosen". In a sense, the axiom of choice "randomly chooses an infinite number of times", so it doesn't make sense to speak of an object "constructed" by the axiom of choice.
In comparison, the proof in the finite dimensional case requires randomly choosing some element of the vector space a couple of times, which "sounds" non-constructive, but if you construct a finite dimensional vector space for me, I can concretely apply the procedure to obtain a base.

>>11469208
>Intuition says its because potential new eigenvectors would be already included due to rotatey nature of complex eigenvalues but im not 100% sure
Ah, so a counting argument, like this?
[math]Ax= \lambda x[/math] implies [math]p(A)x= p( \lambda ) x[/math]. If we have a basis of eigenvectors [math]x_i[/math] of [math]A[/math], we also have a basis of eigenvectors of [math]p(A)[/math], and then it follows trivially.