A9_soln - Math 235 Assignment 9 Solutions 1. Let ~u = (1 +...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 235 Assignment 9 Solutions 1. Let ~u = (1 + i, 2- i ) and ~v = (3- i, 3 + i ). Use the standard inner product on C n to calculate < ~u,~v > , < ~v,~u > , and k ~v k . Solution: We have < ~u,~v > = (1 + i, 2- i ) (3- i, 3 + i ) = (1 + i )(3 + i ) + (2- i )(3- i ) = 2 + 4 i + 5- 5 i = 7- i < ~v,~u > = (3- i, 3 + i ) (1 + i, 2- i ) = (3- i )(1- i ) + (3 + i )(2 + i ) = 2- 4 i + 5 + 5 i = 7 + i k ~v k = p < ~v,~v > = q (3- i, 3 + i ) (3- i, 3 + i ) = p (3- i )(3 + i ) + (3 + i )(3- i ) = 20 2. Determine which of the following matrices is unitary. a) A = (1 + i ) / 7- 5 / 35 (1 + 2 i ) / 7 (3 + i ) / 35 . Solution: Observe that Solution: Let ~u = ((1 + i ) / 7 , (1 + 2 i ) / 7), and ~v = (- 5 / 35 , (3 + i ) / 35). Then we have < ~u,~u > = 1 7 [(1 + i )(1- i ) + (1 + 2 i )(1- 2 i )] = 1 7 [2 + 5] = 1 < ~v,~v > = 1 35 [(- 5)(- 5) + (3 + i )(3- i )] = 1 35 [25 + 10] = 1 < ~u,~v > = 1 7 35 [(1 + i )(- 5) + (1 + 2 i )(3- i )] = 1 7 35 [- 5- 5 i + 5 + 5 i ] = 0 Hence, { ~u,~v } is an orthonormal basis for C 2 and so...
View Full Document

Page1 / 4

A9_soln - Math 235 Assignment 9 Solutions 1. Let ~u = (1 +...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online