16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.2.1

Operators on Inner Product Spaces

Mathematics — Linear Algebra · sesquilinear-forms

Problem

Which of the following functions ff, defined on vectors α=(x1,x2)\alpha = (x_1, x_2) and β=(y1,y2)\beta = (y_1, y_2) in C2C^2, are (sesqui-linear) forms on C2C^2? (a) f(α,β)=1f(\alpha, \beta) = 1. (b) f(α,β)=(x1−yˉ1)2+x2yˉ2f(\alpha, \beta) = (x_1 - \bar{y}_1)^2 + x_2\bar{y}_2. (c) f(α,β)=(x1+yˉ1)2−(x1−yˉ1)2f(\alpha, \beta) = (x_1 + \bar{y}_1)^2 - (x_1 - \bar{y}_1)^2. (d) f(α,β)=x1yˉ2−xˉ2y1f(\alpha, \beta) = x_1\bar{y}_2 - \bar{x}_2 y_1.

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