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# sol64 - Partial Solution Set Leon 6.4 2 compute z w z w and...

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Partial Solution Set, Leon § 6.4 *************************************************************************** 6.4.1a For z = 4 + 2 i 4 i and w = - 2 2 + i , compute k z k , k w k , h z , w i , and h w , z i . Solution : k z k = z H z = 36 = 6, k w k = w H w = 9 = 3, h z , w i = w H z = - 4 + 4 i , and h w , z i = z H w = - 4 - 4 i . 6.4.2b Let z 1 = 1 + i 2 1 - i 2 , and z 2 = i 2 - 1 2 . Write the vector z = 2 + 4 i - 2 i as a linear combination of z 1 and z 2 . Solution : From part (a) of this exercise, we know that { z 1 , z 2 } is an orthonormal set, so we don’t have to work very hard to come up with coefficients c 1 , c 2 such that z = c 1 z 1 + c 2 z 2 . By Theorem 5.5.2 and the definition of the complex inner product, c 1 = h z , z 1 i = 4, and c 2 = h z , z 2 i = 2 2, thus z = 4 z 1 + 2 2 z 2 . 6.4.3 Let { u 1 , u 2 } be an orthonormal basis for C 2 , and let z = (4 + 2 i ) u 1 + (6 - 5 i ) u 2 . (a) What are the values of u H 1 z , z H u 1 , u H 2 z , and z H u 2 ? Solution : u H 1 z = u H 1 ((4 + 2 i ) u 1 + (6 - 5 i ) u 2 ) = (4 + 2 i ) u H 1 u 1 + (6 - 5 i ) u H 1 u 2 = 4 + 2 i.

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sol64 - Partial Solution Set Leon 6.4 2 compute z w z w and...

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