# Linear algebra toughy

• May 4th 2008, 12:28 PM
mathisthebestpuzzle
Linear algebra toughy
Suppose that n is a positive integer. Define T \ $in L(F^n)$ by:

T( $z_1$, ....., $z_n$)=(0, $z_1$,...., $z_n-1$).

Find a formula for T*( $z_1$, ....., $z_n$).

T*is the adjoint of T.
• May 4th 2008, 09:34 PM
CaptainBlack
Quote:

Originally Posted by mathisthebestpuzzle
Suppose that n is a positive integer. Define T \ $in L(F^n)$ by:

T( $z_1$, ....., $z_n$)=(0, $z_1$,...., $z_n-1$).

Find a formula for T*( $z_1$, ....., $z_n$).

T*is the adjoint of T.

The adjoint is defined by the relation:

$\langle T^*(\bold{v}),\bold{w}\rangle = \langle \bold{v},T(\bold{w}) \rangle = v_2w_1+ ... + v_nw_{n-1}$

Therefore:

$T^*(v_1, v_2, ..., v_n)=(v_2, v_3, .., v_n, 0)$

You can get the same result by constructing the matrix if $T$ and taking its transpose.

RonL