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 18E from Chapter 4.6 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 18E
Chapter:
Problem:
In Exercises 17 and 18, A is an m × n matrix. Mark each statement True or False. Justify each answer. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A. b. Row operations preserve the linear dependence relations among the rows of A. c. The dimension of the null space of A is the number of columns of A that are not pivot columns. d. The row space of AT is the same as the column space of A. e. If A and B are row equivalent, then their row spaces are the same.
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 B is any echelon form of A, then the pivot columns of B tell which columns of A will form a basis for the column space of A. The pivot columns of B do not form a basis for A.
The statement is False.
Step-2: (b)
The row operations do not preserve the linear dependence relations. In the row-reduction process, if the interchange of rows occurs, then the row space is changed.
The statement is False.
Step-3: (c)
The dimension of column space of A is given by the number of pivot columns and dimension of null space of A is equal to the number of columns of A not having pivot elements.
The statement is True.
Step-4: (d)
Let a matrix has dimensions $m \times n$. The transpose of the matrix is obtained by interchanging the rows and columns. So, the dimensions of the transpose of the matrix is $n \times m$
Since the rows have been interchanged with the columns during transpose. Hence, the columns space of $A^T$ is same as the row space of A.
The statement is True.
Step-5: (e)
The row equivalent matrices can be obtained from the other by applying elementary row operations.. If two matrices are row equivalent, then they should have the same reduced echelon form.
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 B is any echelon form of A, then the pivot columns of B tell which columns of A will form a basis for the column space of A. The pivot columns of B do not form a basis for A.
The statement is False.
Step-2: (b)
The row operations do not preserve the linear dependence relations. In the row-reduction process, if the interchange of rows occurs, then the row space is changed.
The statement is False.
Step-3: (c)
The dimension of column space of A is given by the number of pivot columns and dimension of null space of A is equal to the number of columns of A not having pivot elements.
The statement is True.
Step-4: (d)
Let a matrix has dimensions $m \times n$. The transpose of the matrix is obtained by interchanging the rows and columns. So, the dimensions of the transpose of the matrix is $n \times m$
Since the rows have been interchanged with the columns during transpose. Hence, the columns space of $A^T$ is same as the row space of A.
The statement is True.
Step-5: (e)
The row equivalent matrices can be obtained from the other by applying elementary row operations.. If two matrices are row equivalent, then they should have the same reduced echelon form.
The statement is True.