Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 24E from Chapter 6.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 24E
Chapter:
Problem:
In Exercises 23 and 24, all vectors are in Rn. Mark each statement True or False. Justify each answer. a. Not every orthogonal set in Rn is linearly independent. d. The orthogonal projection of y onto v is the same as the orthogonal projection of y onto cv whenever c ≠ 0. e. An orthogonal matrix is invertible.
Step-by-Step Solution
Given Information
We are given with some statements that we have to prove whether they are True or False.
Step-1: (a)
If $S = \left\{ \mathbf { u } _ { 1 } , \dots , \mathbf { u } _ { p } \right\}$ is an orthogonal set of nonzero vectors in, $R^n$ then S is linearly independent and hence is a basis for the subspace spanned by S.
So, a condition that an orthogonal set to be linearly independent is that it should be non-zero.
The statement is True.
Step-2:(b)
The given condition is True for an orthogonal set. Not all orthogonal sets are orthogonal.
The statement is False.
Step-3:(c)
If the linear mapping $\mathbf { x } \mapsto A \mathbf { x }$ preserves length, then its definition is: \[ \| A x \| = \| x \| \] If U is an orthonormal columns, then \[ \| U \mathbf { x } \| = \| \mathbf { x } \| \] Which is similar to the condition given above:
The statement is True
Step-4:(d)
The orthogonal projection of y on u is given by: \[ \hat { \mathbf { y } } = \dfrac { \mathbf { y } \cdot \mathbf { u } } { \mathbf { u } \cdot \mathbf { u } } \mathbf { u } \] For an orthogonal projection of y on $cu$, we can replace u by $cu$. . The result still remains same:
The statement is True
Step-5:(e)
An orthogonal matrix is a square invertible matrix such that: \[ U ^ { - 1 } = U ^ { T } \]
The statement is True
We are given with some statements that we have to prove whether they are True or False.
Step-1: (a)
If $S = \left\{ \mathbf { u } _ { 1 } , \dots , \mathbf { u } _ { p } \right\}$ is an orthogonal set of nonzero vectors in, $R^n$ then S is linearly independent and hence is a basis for the subspace spanned by S.
So, a condition that an orthogonal set to be linearly independent is that it should be non-zero.
The statement is True.
Step-2:(b)
The given condition is True for an orthogonal set. Not all orthogonal sets are orthogonal.
The statement is False.
Step-3:(c)
If the linear mapping $\mathbf { x } \mapsto A \mathbf { x }$ preserves length, then its definition is: \[ \| A x \| = \| x \| \] If U is an orthonormal columns, then \[ \| U \mathbf { x } \| = \| \mathbf { x } \| \] Which is similar to the condition given above:
The statement is True
Step-4:(d)
The orthogonal projection of y on u is given by: \[ \hat { \mathbf { y } } = \dfrac { \mathbf { y } \cdot \mathbf { u } } { \mathbf { u } \cdot \mathbf { u } } \mathbf { u } \] For an orthogonal projection of y on $cu$, we can replace u by $cu$. . The result still remains same:
The statement is True
Step-5:(e)
An orthogonal matrix is a square invertible matrix such that: \[ U ^ { - 1 } = U ^ { T } \]
The statement is True