Suppose n is a positive integer.Define T∈ L(Fn) by T(z1, z2,....., zn)=(0, z1,..., zn-1). Find a formula for T*(z1, z2,....., zn).
Fix(z1...zn)ϵFn.Then for every(w1...wn)ϵFn,we have((w1...wn),T∗(z1...zn))=(T(w1...wn),(z1...zn))=((0,w1...wn−1),(z1...zn))=w1z2ˉ+...+wn−1znˉ=((w1...wn),(z2....zn,0))ThusT∗(z1...zn)=(z2...zn,0)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)Fix(z1...zn)ϵFn.Then for every(w1...wn)ϵFn,we have((w1...wn),T∗(z1...zn))=(T(w1...wn),(z1...zn))=((0,w1...wn−1),(z1...zn))=w1z2ˉ+...+wn−1znˉ=((w1...wn),(z2....zn,0))ThusT∗(z1...zn)=(z2...zn,0)
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