Structure Of Linear Operators 2

Question 1

If Vi <= V with V = V1+o…+o Vk, and phi = phi1 +o …+o phik, what are ker(phi) and im(phi)

Answer 1

Ker(phi) = ker(phi1) +o…+o ker(phik)
Im(phi) = im(phi1) +o … +o im(phik)

Question 2

If Vi <= V with V = V1+o…+o Vk, and phi = phi1 +o …+o phik what are p(phi) and characteristic polynomial of phi

Answer 2

P(phi) = p(phi1) +o…+o p(phik)
Characteristic polynomial of phi is PI I = 1 to k characteristic polynomials of phi(i)

Question 3

When is linear operator phi diagonalisable

Answer 3

Linear operator phi is diagonalisable iff V = direct sum i=1 to k Ephi(Li) where Li are distinct eigenvalues of phi, E is eigenspace of phi
To summarise, phi(i) = L(I) * id(Vi)

Question 4

If phi is a linear operator, what is ker(phi^k)

Answer 4

Ker(phi^k) <= ker(phi^k+1) and ker(phi^k) = ker(phi^k+1) implies ker(phi^k) = ker(phi^k+n) all kernels are same.

Question 5

If phi is a linear operator, what is im(phi^k)

Answer 5

Im(phi^k) >= im(phi^k+1) if im(phi^k) = im(phi^k+1) then im(phi^k) = im(phi^k+n)

Question 6

If dim V = n, what are ker(phi^n) and im(phi^n)

Answer 6

Ker(phi^n) = ker(phi^n+k) and im(phi^n) = im(phi^n+k)

Question 7

If dim V = n, what is V

Answer 7

If dim v = n, then V = ker(phi^n) +o im(phi^n)

Question 8

What is generalised eigenvector of phi with eigenvalue L

Answer 8

Generalised eigenvector of phi with eigenvalue L is:
(Phi - L(idV))^k (v) = 0

Question 9

What is generalised eigenspace

Answer 9

Generalised eigenspace is set of all general eigenvectors.
G(L) = (v in V | (phi - L(idV)^k (v) = 0 k in N) where L is all E.values and k is dim V
= U ker(phi - Lidv)^k

Question 10

What is relation between eigenspace and generalised eigenspace

Answer 10

Relation between eigenspace and generalised eigenspace is :
Ephi(L) <= Gphi(L) <= V subspaces

Question 11

What is Gphi(L1) N G(phi)(L2)

Answer 11

Gphi(L1) N Gphi(L2) = 0 as all eigenvectors associated to each elvalue are distinct

Question 12

What is Jordan decomposition theorem

Answer 12

Jordan decomposition theorem is:
If f V is finite dimensional, then V = direct sum i=1 to k Gphi(Li) where Li are distinct eigenvalues of phi E L(V)
Dim of each is algebraic multiplicity

Question 13

When is a linear operator phi nilpotent

Answer 13

Linear operator phi is nilpotent if phi^k = 0 k in N, if V is finite dimensional, k = dim V

Question 14

How can V be written as a direct sum using eigenspace

Answer 14

V can be written as a direct sum using eigenspace by:
V = direct sum i =1 to k G(phi)(Li) where the dimension is of each eigenspace is a.n of E.vector

Question 15

What is characteristic polynomial

Answer 15

Characteristic polynomial is:
+- PI i =1 to k (x-Li)^Mi where mi = a.m(Li) and Li s are E.values