Answer to Question #220127 in Linear Algebra for Sabelo

Question

Suppose n is a positive integer.Define T∈ L(Fⁿ) by T(z₁, z₂,....., z_n)=(0, z₁,..., z_n-1). Find a formula for T*(z₁, z₂,....., z_n).

Answer 1

$Fix (z_1... z_n) ϵ F^n. Then \space for \space every (w_1...w_n) ϵ F^n, we \space have\\ ((w_1...w_n), T ^∗(z_1... z_n)) = (T (w_1... w_n), (z_1... z_n))\\ = ((0,w_1... w_{n−1}), (z_1... z_n))\\ = w_1\bar{z_2} + ... + w_{n−1} \bar{z_n}\\ = ((w_1... w_n), (z_2.... z_n, 0))\\ Thus\\ T ∗(z_1...z_n) = (z_2... z_n, 0)$