There are five questions. The questions are the content in the chapter 8 and 10 in the book “Linear done right”. Please answer it as specify as possible

1. Define a linear operator T on R3 such that T is nilpotent and satisfies

(C) = 0

T

(8) – 3

2. Find all possible Jordan canonical forms of 4 x 4 complex matrices which satisfy the

following three conditions simultaneously:

• A is not diagonalizable;

• A has characteristic polynomial (1 – 5)²(x – 1)?;

• A satisfies the equation (A – 51)(A – 1) = 0.

3. Let V = Rº be the inner product space equipped with the inner product

12

Y2

2:01:22 +242 +2241 + yıy2.

Let T be the linear operator defined by

T

(

©) = (2+2)

Answer the following two questions with justifications.

(a) Is T diagonalizable?

(b) Does there exist an orthonormal basis of V consisting of eigenvectors of T? (Or-

thonormal with respect to the above inner product.)

4. Let V be the vector space of nxn complex matrices. Let A be a fixed matrix. Define

a map T :V + V by T(B) = AB – BA.

(a) Verify that T is a linear operator on V.

(b) Prove that if A is a nilpotent matrix, then T is a nilpotent operator.

0

5. (a) Let A=

be a diagonal complex matrix and f(2) be a polynomial

0 An

with coefficients in C. Describe f(A).

(b) Let T be a normal operator on an n-dimensional complex inner product space.

Prove that there is a polynomial f, with complex coefficients, such that

f(T) = T”

(You may use without proof the following fact: Given any two n-tuples of complex

numbers (41, …, In) and (H1, …,Hr) there is a polynomial f(1), with complex

coefficients, such that f(1) = M1, …, f (An) = for.)