Kunze Linear Algebra · Problem 14

Inner Product Spaces

Mathematics — Methods · rigid-motions

Problem

V

is an inner product space, a rigid motion is any function

T

from

V

into

V

(not necessarily linear) such that

||T\alpha - T\beta|| = ||\alpha - \beta||

for all

\alpha

\beta

V

. One example of a rigid motion is a linear unitary operator. Another example is translation by a fixed vector

\gamma

T_\gamma(\alpha) = \alpha + \gamma

(a) Let

V

R^2

with the standard inner product. Suppose

T

is a rigid motion of

V

and that

T(0) = 0

. Prove that

T

is linear and a unitary operator. (b) Use the result of part (a) to prove that every rigid motion of

R^2

is composed of a translation, followed by a unitary operator. (c) Now show that a rigid motion of

R^2

is either a translation followed by a rotation, or a translation followed by a reflection followed by a rotation.