16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.4.12

Linear Transformations

Mathematics — Linear Algebra · nilpotent-operator-basis

Problem

Let

V

be an

n

-dimensional vector space over the field

F

, and let

\mathfrak{B} = \{\alpha_1, \ldots, \alpha_n\}

be an ordered basis for

V

. (a) According to Theorem 1, there is a unique linear operator

T

V

such that

T\alpha_j = \alpha_{j+1}, \quad j = 1, \ldots, n-1, \qquad T\alpha_n = 0.

What is the matrix

A

T

in the ordered basis

\mathfrak{B}

? (b) Prove that

T^n = 0

but

T^{n-1} \neq 0

. (c) Let

S

be any linear operator on

V

such that

S^n = 0

but

S^{n-1} \neq 0

. Prove that there is an ordered basis

\mathfrak{B}'

for

V

such that the matrix of

S

in the ordered basis

\mathfrak{B}'

is the matrix

A

of part (a). (d) Prove that if

M

and

N

are

n \times n

matrices over

F

such that

M^n = N^n = 0

but

M^{n-1} \neq 0 \neq N^{n-1}

, then

M

and

N

are similar.