16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.4.15

Bilinear Forms

Mathematics — Linear Algebra · determinant-bilinear-form-c2

Problem

Let ff be the bilinear form on C2C^2 defined by f((x1,x2),(y1,y2))=x1y2−x2y1.f((x_1, x_2), (y_1, y_2)) = x_1 y_2 - x_2 y_1. Show that (a) if TT is a linear operator on C2C^2, then f(Tα,Tβ)=(det⁡T)f(α,β)f(T\alpha, T\beta) = (\det T) f(\alpha, \beta) for all α,β\alpha, \beta in C2C^2. (b) TT preserves ff if and only if det⁡T=+1\det T = +1. (c) What does (b) say about the group of 2×22 \times 2 matrices MM such that MtAM=AM^t A M = A where A=[01−10]?A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}?

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