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Math Amateur

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Some terminology from Grove and Benson:

V is a real Euclidean vector space

A transformation of V is understood to be a linear transformation

The group of all orthogonal transformations of V will be denoted O(V)

Then in chapter 2, Grove and Benson write the following:

If T [itex]\in[/itex] O(V), then T is completely determined by its action on the basis vectors [itex]e_1[/itex] = (1,0) and [itex]e_2[/itex] = (0,1).

If T[itex]e_1 = ( \mu , \nu )[/itex], then [itex]{\mu}^2 + {\nu}^2 = 1[/itex] and [itex]T e_2 = \pm ( - \nu , \mu) [/itex]

Can someone please help by proving why the last statement is true?

Peter