Suppose T : Rn Rn has the following properties: (i) T (0) = 0; (ii) T preserves distance

QUESTION:

Suppose T : Rn Rn has the following properties: (i) T (0) = 0; (ii) T preserves distance (i.e., _T (x) T (y)_ = _x y_ for all x, y Rn). a. Prove that T (x) T (y) = x y for all x, y Rn. b. If {e1, . . . , en} is the standard basis, letT (ei) = vi . Prove that T __n i=1 xiei = _n i=1 xivi . c. Deduce from part b that T is a linear transformation. d. Prove that the standard matrix for T is orthogonal.