Consider the set O2 = {A GL2; such that (Au, Av) = (u, v) for...

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Consider the set O2 = {A € GL2; such that (Au, Av) = (u, v) for every u, v E R²}, i.e., the set of all 2 x 2 real matrices that preserve the inner product. Show that the triple (O2, , I2) is a subgroup of GL2(R), where denotes the standard multiplication of matrices. This group is called the real orthogonal group of order two. Consider the set O2 = {A € GL2; such that (Au, Av) = (u, v) for every u, v E R²}, i.e., the set of all 2 x 2 real matrices that preserve the inner product. Show that the triple (O2, , I2) is a subgroup of GL2(R), where denotes the standard multiplication of matrices. This group is called the real orthogonal group of order two.