Math Homework

 

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Prob 1.    Let V   be a complex n-dimensional space and let T    (V ) be such that null T ⁿ⁻3 = null T ⁿ⁻2. How many distinct eigenvalues can T have?

Prob 2.           Let V be a complex finite-dimensional vector space and let T     (V ) have eigenvalues  1, 0, 1. Given the dimensions of the corresponding nullspaces below, determine the Jordan normal form of T

Prob 3. Let T ∈ L(P3(C)) be the operator

T : f (x) ›→ f (x − 1) + x3f ^jjj(x)/3.

Find the Jordan normal form and a Jordan basis for T .

Prob 4.    Let V be a complex (finite-dimensional) vector space and let T ∈ L(V ). Prove that there exist operators D and N in L(V ) such that T = D + N , D is diagonalizable, N is nilpotent, and DN = ND.

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Prob 5.     Suppose that V is a complex vector space of dimension n.  Let T    (V ) be invertible.  Let p

denote the characteristic polynomial of T and let q denote the characteristic polynomial of T ⁻1. Prove that 

Prob 6. Suppose the Jordan form of an operator T ∈ L(V ) consists of Jordan blocks of sizes 3 × 3, 4 × 4,

1 × 1, 5 × 5, 2 × 2, corresponding to eigenvalues λ1, λ2, λ3, λ2, λ1, respectively. Assuming that λ_i  i ƒ= j, find the minimal and the characteristic polynomial of T .