VII. Prove the following statements. 1. Let B be an orthonormal basis of a finite-dimensional inner product space V. Suppose that the a1 02 coordinate vector [v]B = Then |v|a₁|²+|a₂|² + = ·+|an|². 2. If A is similar to a diagonal matrix D then A is similar to AT. 3. Let A be an n x n matrix with eigenvalues A₁ and A2, where A₁ A2. If Sx, is the eigenspace associated with Ai, i = 1,2, then Sx, Sx₂ = {0}.
VII. Prove the following statements. 1. Let B be an orthonormal basis of a finite-dimensional inner product space V. Suppose that the a1 02 coordinate vector [v]B = Then |v|a₁|²+|a₂|² + = ·+|an|². 2. If A is similar to a diagonal matrix D then A is similar to AT. 3. Let A be an n x n matrix with eigenvalues A₁ and A2, where A₁ A2. If Sx, is the eigenspace associated with Ai, i = 1,2, then Sx, Sx₂ = {0}.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
prove no.1
![VII. Prove the following statements.
1. Let B be an orthonormal basis of a finite-dimensional inner product space V. Suppose that the
a1
a2
coordinate vector [v]B =
Then |v|a₁|²+|a₂|² +
·+|an|².
2. If A is similar to a diagonal matrix D then A is similar to AT.
3. Let A be an n x n matrix with eigenvalues A₁ and A2, where A₁ A2. If Sx, is the eigenspace
associated with Ai, i = 1,2, then Sx₁ Sx₂ = {0}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1be657f-6b04-4fb6-883d-75c0198282db%2Fb1546277-e4bf-46d8-9c99-45bb44448a23%2Fndgxoi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:VII. Prove the following statements.
1. Let B be an orthonormal basis of a finite-dimensional inner product space V. Suppose that the
a1
a2
coordinate vector [v]B =
Then |v|a₁|²+|a₂|² +
·+|an|².
2. If A is similar to a diagonal matrix D then A is similar to AT.
3. Let A be an n x n matrix with eigenvalues A₁ and A2, where A₁ A2. If Sx, is the eigenspace
associated with Ai, i = 1,2, then Sx₁ Sx₂ = {0}.
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