Linear Algebra, Theory And Applications, 2012a

242 LINEAR TRANSFORMATIONS
9.5 Exercises
1. If A, B, and C are each n × n matrices and ABC is invertible, why are each of A, B,
and C invertible?
2. Give an example of a 3 × 2 matrix with the property that the linear transformation
determined by this matrix is one to one but not onto.
3. Explain why Ax = 0 always has a solution whenever A is a linear transformation.
4. Review problem: Suppose det (A − λI) =0. Show using Theorem 3.1.15 there exists
x ≠ 0 such that (A − λI) x = 0.
5. How does the minimal polynomial of an algebraic number relate to the minimal polynomial
of a linear transformation? Can an algebraic number be thought of as a linear
transformation? How?
6. Recall the fact from algebra that if p (λ) andq (λ) are polynomials, then there exists
l (λ) , a polynomial such that
q (λ) =p (λ) l (λ)+r (λ)
where the degree of r (λ) islessthanthedegreeofp (λ) orelser (λ) =0. With this in
mind, why must the minimal polynomial always divide the characteristic polynomial?
That is, why does there always exist a polynomial l (λ) such that p (λ) l (λ) =q (λ)?
Can you give conditions which imply the minimal polynomial equals the characteristic
polynomial? Go ahead and use the Cayley Hamilton theorem.
7. In the following examples, a linear transformation, T is given by specifying its action
on a basis β. Find its matrix with respect to this basis.
( ) ( ) ( ) ( ) ( )
1 1 −1 −1 −1
(a) T =2 +1 ,T =
2 2 1 1 1
( ) ( ) ( ) ( ) ( )
0 0 −1 −1 0
(b) T =2 +1 ,T =
1 1 1 1 1
( ) ( ) ( ) ( ) ( ) ( )
1 1 1 1 1 1
(c) T =2 +1 ,T =1 −
0 2 0 2 0 2
8. Let β = {u 1 , ··· , u n } be a basis for F n and let T : F n → F n be defined as follows.
( n
)
∑
n∑
T a k u k = a k b k u k
k=1
First show that T is a linear transformation. Next show that the matrix of T with
respect to this basis, [T ] β
is
⎛
⎞
b 1
⎜
⎝
. ..
⎟
⎠
b n
k=1
Show that the above definition is equivalent to simply specifying T on the basis vectors
of β by
T (u k )=b k u k .