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 22E from Chapter 4.3 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 22E
Chapter:
Problem:
In Exercises 21 and 22, mark each statement True or False. Justify each answer. a. A linearly independent set in a subspace H is a basis for H. b. If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V. c. A basis is a linearly independent set that is as large as possible. d. The standard method for producing a spanning set for Nul A, described in Section 4.2, sometimes fails to produce a basis for Nul A. e. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
Step-by-Step Solution
Given Information
We are given with a set statements that we have to prove whether the statements are true or false.
Step 1: (a)
A set is called a basis for the space only if the set is linearly independent ans spans the space. The statement does not include the term space. Hence
The statement is False.
Step 2: (a)
Suppose we are given with a set H of vectors such that the vectors are linearly independent then the vectors in the set forms a basis for the H.
However, if the set is not linearly independent, then we write one of the vectors in terms of other vectors. Then we remove that vector from the set and if the remaining vectors are linearly independent then it forms as basis for H.
The statement is True.
Step 3: (c)
Suppose we are given with a set H of vectors such that the vectors are linearly independent then the vectors in the set forms a basis for the H.
However, if the set is not linearly independent, then we write one of the vectors in terms of other vectors. Then we remove that vector from the set and if the remaining vectors are linearly independent then it forms as basis for H. Thus the set is as large as possible.
The statement is True.
Step 4: (d)
The standard method for finding basis for Nul A produces linearly independent If the null space of a matrix A contains nonzero vectors.
The statement is False.
Step 5: (e)
The pivot columns of B only tell that which columns of A form a basis for Col A. The columns of B do not form the basis for Col A
The statement is False.
We are given with a set statements that we have to prove whether the statements are true or false.
Step 1: (a)
A set is called a basis for the space only if the set is linearly independent ans spans the space. The statement does not include the term space. Hence
The statement is False.
Step 2: (a)
Suppose we are given with a set H of vectors such that the vectors are linearly independent then the vectors in the set forms a basis for the H.
However, if the set is not linearly independent, then we write one of the vectors in terms of other vectors. Then we remove that vector from the set and if the remaining vectors are linearly independent then it forms as basis for H.
The statement is True.
Step 3: (c)
Suppose we are given with a set H of vectors such that the vectors are linearly independent then the vectors in the set forms a basis for the H.
However, if the set is not linearly independent, then we write one of the vectors in terms of other vectors. Then we remove that vector from the set and if the remaining vectors are linearly independent then it forms as basis for H. Thus the set is as large as possible.
The statement is True.
Step 4: (d)
The standard method for finding basis for Nul A produces linearly independent If the null space of a matrix A contains nonzero vectors.
The statement is False.
Step 5: (e)
The pivot columns of B only tell that which columns of A form a basis for Col A. The columns of B do not form the basis for Col A
The statement is False.