Originally Posted by

**Nona** Hello,

i would like help with this:

Assume that T : V ....> V is linear and that n and k are positive integers and that a_0,a_1,a_2,.....,a_n are scalars. Let U = a_0IV + a_1T + a_2T^2 + + a_nT^n. Show that T^kU = UT^k.

Use from 1 to 6:

(1) Since

L(V, V ) is a vector space all of the vector space properties hold for addition and scalar multiplication.

(2)

T(U1U2) = (TU1)U2.

(3) )

T(U1 + U2) = (TU1) + (TU2).

(4) (U1 + U2)T = U1T + U2T.

(5) For any non-negative integers

i and j, T^iT^j = T^i+j .

(6) For any scalars c and d, (cT )(dU1) = (cd)TU1.

Thank you

This is trivial using the given definitions. For example, with k = 2 we get:

$\displaystyle T^2U=T^2(a_0I+a_1T+...+a_nT^n)=T\left[T\left(\sum\limits_{i=0}^na_iT^i\right)\right]=$ $\displaystyle T\left(\sum\limits_{i=0}^na_iT^{i+1}\right)=\sum\l imits_{i=0}^na_iT^{i+2}$

$\displaystyle UT^2=(UT)T=\left[\left(\sum\limits_{i=0}^na_iT^{i}\right)T\right]T$ $\displaystyle =\left[\sum\limits_{i=0}^na_iT^{i+1}\right]T=\sum\limits_{i=0}^na_iT^{i+2}$

Now just apply a simple induction here and you're done.

Tonio