(1) [5 points] For x, y € C&quot;, recall that n = (1) ...

Transcribed Text

(1) [5 points] For x, y € C", recall that n = (1) k=1 If x, y, 2 E Cn and a, B €.C, prove that (ax + By,z) = a(x,2) + B(y,2). (2) [5 points] For x E C", recall that n == (2) k=1 If x € Cn and a € C, use properties of the modulus to prove that (3) [5 points] If p(z) = 22 + a12 + ao € P2(C) and C: 10-11 0 ao - 1 ai , prove that pc = p. (4) [10 points] A matrix A E Mn (C) is called negative definite if 2* Az < 0, Vz € Cn, 2 # 0. If A is negative definite and \ € spec(A), prove that \ < 0. Hint: Compute U* Av, in which U is an eigenvector of A associated with 1.

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