Orthogonal linear transformation

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Reference no: EM13124041

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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.

Reference no: EM13124041

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