Lemma 6.7.1 - Unit 4 - Canonical Forms: Rational Canonical Form

[poojar]

Theorem 6.7.3 explained with corollary

[caramel_nilavu]

V is Cyclic relative to T is nothing but V is cyclic with respect to T we know that T is in A(V) and V is vector space. Whenever we say V is cyclic means V contain an element that generates all the other elements in that vector space(ie. The whole vector space is generated by 1 particular element in V) with the help of something. Here they have said V is cyclic "with respect to T" this tells us that T helps that particular element to generate the whole vector space. As we are talking about polynomial in this place T acts as polynomial. To write these above things in mathematical form. Let me name that particular element as "w" in V And let me denote polynomial of T to be f(T) and let me denote elements of V be small " v " now the mathematical form of the above things will be * v = w(f(T)) * this mean that if we take that particular element in V and multiply it with any f(T) we will get the other elements in V .
That's it