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**Unformatted text preview: **N ( A ) ≥ m 2 or dim N ( B ) ≥ m 2 . Problem #4. (30 pts) a) Suppose that A,B ∈ C m × m are unitary matrices. Verify that A * and AB are also unitary. (Hint: Use the algebraic properties of the adjoint) b) Let v 1 = 1 5 3-4 , v 2 = 1 5 4 3 , v 3 = 1 ∈ R 3 Verify that { v 1 , v 2 , v 3 } is an orthonormal basis of R 3 . Justify your answer. c) Let w 1 = √ 2 2 1 1 , w 2 = -1 , w 3 = √ 2 2 -1 1 ∈ R 3 be a set of orthonormal vectors. Determine the orthogonal matrix U ∈ R 3 × 3 so that U v i = w i for i = 1 , 2 , 3 here the v i are given in b). (Hint: Use part a) )...

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