Theorem 12.8 Let A be an n × n matrix. The following statements are equivalent. 1. The columns of the matrix A form an orthonormal set. 2. A−1 = At . 3. For vectors x and y, ⟨Ax, Ay⟩ = ⟨x, y⟩. 4. For vectors x and y, ∥Ax − Ay∥ = ∥x − y∥. 5. For any vector x, ∥Ax∥ = ∥x∥.

Theorem 12.8 Let A be an n × n matrix. The following statements are equivalent. 1. The columns of the matrix A form an orthonormal set. 2. A−1 = At . 3. For vectors x and y, ⟨Ax, Ay⟩ = ⟨x, y⟩. 4. For vectors x and y, ∥Ax − Ay∥ = ∥x − y∥. 5. For any vector x, ∥Ax∥ = ∥x∥.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.6: The Algebra Of Matrices

Problem 14E

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Question

Theorem 12.8 Let A be an n × n matrix. The following statements are equivalent.
1. The columns of the matrix A form an orthonormal set.
2. A−1 = At
.
3. For vectors x and y, ⟨Ax, Ay⟩ = ⟨x, y⟩.
4. For vectors x and y, ∥Ax − Ay∥ = ∥x − y∥.
5. For any vector x, ∥Ax∥ = ∥x∥.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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