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Post by omnipotentvoid on May 9, 2017 18:49:35 GMT
While doing some maths exercises, I stumbled upon this little problem and thought I'd share it: Let V be a finite-dimensional k-vectorspace, F be a vector space endomorphism and v an element of V so that: F^i(v) = (F◦F◦...◦F)(V), especially F^0(v)=v a) Show that: there exist a k in the natural numbers so that the vectors v,F(v),F^2(v),...,F^k-1(V) are linearly independent, but the vectors v,F(v),F^2(v),...,F^k-1(V),F^k(V) are linearly dependant. b) Deduce that the subspace U = span(v,F(v),...,F^k-1(v)) has the the base (v,F(v),...,F^k-1(v)) c) Show that F(U) is contained within U and represent F restricted to U in respect to the base (v,F(v),...,F^k-1(v)) as a matrix
Have fun . Wikipedia articles to the relevant structures: fieldsvector spacesendomorphism
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