7 Let O V be the set of all orthogonal transformations on V

7. Let O( V) be the set of all orthogonal transformations on V. Prove O( V) is a group with respect to the operation of multiplication.

Solution

Let O(V) be the set of all orthogonal transformations on V.

The determinant of an orthogonal matrix is 1 or -1.

As it is non singular, every matrix A has an inverse.

1) For two orthogonal matrices A and B AB is also orthogonal and closed under multiplication

2) As matrix multiplication is associative Ii axiom is also satisfied by O(V)

3) Identity matrix is also orthogonal with det =1 and belongs to O(V)
4) Every matrix A has inverse = A transpose which is again orthogonal.

Hence O(V) form a group under multiplication.