Kunze Linear Algebra · Problem 3.4.4

Linear Transformations

Mathematics — Methods · cayley-hamilton-two-dimensional

Problem

Let

V

be a two-dimensional vector space over the field

F

, and let

\mathfrak{B}

be an ordered basis for

V

. If

T

is a linear operator on

V

and

[T]_{\mathfrak{B}} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

prove that

T^2 - (a + d)T + (ad - bc)I = 0