Please see attachment for solution.
A linear transformation T: V -> V is said to be orthogonal iif ||T(x)|| = ||x|| for ever x E V.
(a) Prove that T is orthogonal iff < T(x), T(y) >=< x,y > for all x,y E V.
(b) Prove T is orthogonal iff T maps an orthonormal basis {x_1, x_2, ..., x_n} to an orthonormal basis {T(x_1), T(x_2), ..., T(x_n)}.
(c) Let alpha = {x_1, x_2, ..., x_n} be an arbitrary orthonormal basis for V. Prove that T is orthogonal iff [T]_alphaalpha is an orthogonal matrix.