16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.6.1

Linear Transformations

Mathematics — Linear Algebra · annihilator-hyperplane

Problem

Let nn be a positive integer and FF a field. Let WW be the set of all vectors (x1,…,xn)(x_1, \ldots, x_n) in FnF^n such that x1+⋯+xn=0x_1 + \cdots + x_n = 0. (a) Prove that W0W^0 consists of all linear functionals ff of the form f(x1,…,xn)=c∑j=1nxj.f(x_1, \ldots, x_n) = c \sum_{j=1}^{n} x_j. (b) Show that the dual space W∗W^* of WW can be `naturally' identified with the linear functionals f(x1,…,xn)=c1x1+⋯+cnxnf(x_1, \ldots, x_n) = c_1 x_1 + \cdots + c_n x_n on FnF^n which satisfy c1+⋯+cn=0c_1 + \cdots + c_n = 0.

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