Kunze Linear Algebra · Problem 3.4.10

Linear Transformations

Mathematics — Methods · idempotent-operator-classification

Problem

We have seen that the linear operator

T

R^2

defined by

T(x_1, x_2) = (x_1, 0)

is represented in the standard ordered basis by the matrix

A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}.

This operator satisfies

T^2 = T

. Prove that if

S

is a linear operator on

R^2

such that

S^2 = S

, then

S = 0

, or

S = I

, or there is an ordered basis

\mathfrak{B}

for

R^2

such that

[S]_{\mathfrak{B}} = A

(above).