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Problem 2 Suppose that U and V are unitary. For each of the following statements, either prove or give a counterexample that proves the statement false. 1. The matrix product UV is always unitary. 2. The matrix sum U + V is always unitary. Hint: You’ll have to use the definition of unitary to make any progress. While no number of examples is sufficient to prove a statement, it takes only a single counterexample to disprove a statement; look for a simple one. Problem 3 Prove that the operation ⟨·, ·⟩ defined on the complex vector space C2 by ⟨x, y⟩ = x*y meets all the requirements to be an inner product

          Problem 2 Suppose that U and V are unitary. For each of the following statements, either prove
or give a counterexample that proves the statement false.
1. The matrix product UV is always unitary.
2. The matrix sum U + V is always unitary.
Hint: You’ll have to use the definition of unitary to make any progress. While no number of
examples is sufficient to prove a statement, it takes only a single counterexample to disprove a
statement; look for a simple one.
Problem 3 Prove that the operation ⟨·, ·⟩ defined on the complex vector space C2 by
⟨x, y⟩ = x*y
meets all the requirements to be an inner product

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Added by Jasmine M.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Adi S

11/04/2023

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Problem 2 Suppose that U and V are unitary. For each of the following statements, either prove or give a counterexample that proves the statement false. 1. The matrix product UV is always unitary. 2. The matrix sum U + V is always unitary. Hint: You’ll have to use the definition of unitary to make any progress. While no number of examples is sufficient to prove a statement, it takes only a single counterexample to disprove a statement; look for a simple one. Problem 3 Prove that the operation ⟨·, ·⟩ defined on the complex vector space C2 by ⟨x, y⟩ = x*y meets all the requirements to be an inner product

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00:01 So, according to our first part, we need to write here, suppose u is unitary.

00:05 So, we have to write here that is pen of u ,n, u ,u ,w that is u ,v ,w.

00:13 Every v ,w belongs to cn.

00:22 So, from implies that we need to write here that is, that is pen of u ,v ,u ,u ,w that is equal to we need to write v ,w to write here that is v ,u ,u -in and this is equal to we need to write here 0 for every v ,w below.

00:58 So, let's say v is equal to we have to write u transpose u -in ,w.

01:09 So, from here the absolute value we need to write here that is ut ,u -in ,w that is that whole square is equal to 0.

01:25 So, therefore we need to write that is ut -in is equal to 0.

01:34 As we can see that, we can see that say that w is not equal to 0, w is not equal to 0 belongs to the power n.

01:46 So, from here we need to write ut -in.

01:53 So, from here we can write that is let's say let u transpose v that is determinant of in.

02:04 So, implies that we have to write that is determinant of a determinant of a u ,v square that is equal to 1, u square that is again u.

02:16 So, determinant of a u square that is equal to 1.

02:20 So, that is determinant of u is equal to plus minus 1 and since we have to write determinant of a u transpose is equal to determinant of u.

02:31 So, u is invertible...

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