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Let S"X" denote the vector space of symmetric n x n matrices. Define f: Roxn > Sax by f(A) := AT A and define O(n, R) := (In) = (0 ER"x" : OTO = In). (a) What is the dimension of S"xn? (Write down a basis.) (b) Show that Of(O) has maximal rank at all matrices O E O(n, R). Remark. This shows that the orthogonal group O(n, R) is a submanifold of Roxn ~ R""