Linear Algebra II (Fall, 2026)
Homework 4
Due date: October 7 (Wednesday), 2026
> Let T : V → V be a linear operator, where V is a finite dimensional vector space.
1. Let λ be an eigenvalue of T . Using the definition of generalized eigenspace, prove
that Kλ is a subspace of V .
2. Let λ and µ be distinct eigenvalues of T . Prove that Kλ ∩ Kµ = ∅.
3. Suppose that char(T ) splits. Let λ1 , λ2 , . . . , λk be the distinct eigenvalues of T .
Prove that T is diagonalizable if and only if N (T − λi IV ) = N ((T − λi IV )2 ) for every
1 ≤ i ≤ k.
7
4. Let β = {v1 , v2 , . . . , v7 } be an ordered
let T : R7 → R7 be a
 basis for R , and 
3 1 0 0 0 0 0
0 3 1 0 0 0 0


0 0 3 0 0 0 0



0
0
0
3
1
0
0
linear transformation such that [T ]β = 


0 0 0 0 3 0 0


0 0 0 0 0 4 1
0 0 0 0 0 0 4
(1) Find char(T ), and find the eigenvalues of T .
(2) For each eigenvalue λ of T , find the dimension of Kλ .
(3) For each i = 1, 2, . . . , 7, write T (vi ) as a linear combination of vectors in β.
(4) How many cycles of generalized eigenvectors does β consist of? Write each cycle
explicitely from the initial vector to the terminal vector.
5. Suppose that char(T ) splits, and that λ1 , λ2 are the distinct eigenvalues of T .
Prove that for every vector x ∈ V , x = v1 + v2 for some v1 ∈ Kλ1 and v2 ∈ Kλ2 .