Linear Operators/matrices

Question 1

What is a linear operator/endomorphism and how is set denoted

Answer 1

Linear operator is a linear map phi V to V
Set is denoted by L(V)

Question 2

How can we write phi(v) as a matrix v is basis vector

Answer 2

We can write phi(v) as a matrix by:
Sum i = 1 to n Aij*Vi

Question 3

How do we define multiplication on L(V)

Answer 3

Define multiplication on L(V) by composition

Question 4

What is P(phi) if phi is a linear operator

Answer 4

P(phi) is sum k in N, Ak phi^k (coeffs)

Question 5

What is P(A) if A is a matrix

Answer 5

P(A) is sum k in N ak * A^k (coeffs* matrix)

Question 6

What are (p+q)(phi) and (pq)(phi)

Answer 6

(P+q)(phi) = p(phi) + q(phi) and (pq)(phi) = p(phi)q(phi)

Question 7

What is (p+q)(A) and (pq)(A)

Answer 7

(P+q)(A) = p(A) + q(A) and (pq)(A) = p(A)q(A)

Question 8

For a matrix A and linear operator phi, what polynomial exists

Answer 8

For a matrix A, monic polynomial exists such that P(A) = 0, polynomial s.t P(A)
For linear operator phi, exists monic polynomial s,t p(phi) = 0

Question 9

What is a minimum polynomial for L.O phi

Answer 9

Minimum polynomial for L.O phi is a monic polynomial of minimum degree with p(phi) = 0

Question 10

Why is minimum polynomial important

Answer 10

Minimum polynomials are important because p(phi) = 0 iff m(phi) (min polynomial of phi) divided p, p = sm(phi)

Question 11

What all linear operators and matrices have on finite dim V space

Answer 11

ALl linear operators and matrices have a min poly on a V space m(A), m(phi)