Let V be a finite-dimensional vector space over a field F and T ∈ L (V). Suppose that the characteristic polynomial of T , say ft (A), has the factorization ft(A) =g1(A)g2(A) over F where g1and g2 are relatively prime polynomials.Then show that both Ker(g1(T)) and Ker(g2(T)) are invariant under T and V=Ker(g1(T)) ⊕Ker(g2(T)).

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.