MathDB

1. Let

a_{1}, a_{2}, \dots , a_{n}

and

b_{1}, b_{2}, \dots , b_{m}

n+m

unit vectors in the

r

-dimensional Euclidean space

E_{r} (n,m \leq r)

; let

a_{1}, a_{2}, \dots , a_{n}

as well as

b_{1}, b_{2}, \dots , b_{m}

be mutually orthogonal. For any vector

x \in E_{r}

, consider

Tx= \sum_{i=1}^{n}\sum_{k=1}^{m}(x,a_{i})(a_{i},b_{k})b_{k}

(

(a,b)

denotes the scalar product of

a

and

b

). Show that the sequence

(T^{k}x)^{\infty}_{ k =0}

, where

T^{0} x= x

and

T^{k} x = T(T^{k-1}x)

, is convergent and give a geometrical characterization of how the limit depends on

x

. (S. 14)

Problem Statement