Linear algebra toughy

**mathisthebestpuzzle** · May 4th 2008, 12:28 PM

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.

**CaptainBlack** · May 4th 2008, 09:34 PM

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

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