Book edition 5th

Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald

Pages 483 pages

ISBN 978-03219822384

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Short Answer

In Exercise 18, A is an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.
18. 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 \({A^T}\) is the same as the column space of A.
e. If A and B are row equivalent, then their row spaces are the same.**

The statement is false.
The statement is false
The statement is true.
The statement is true.
The statement is true.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the fact of pivot columns

(a)

Note that the pivot columns of matrix A only form a basis for Col A.

Hence, the given statement is false.

Step 2: Use the properties of a basis

(b)

The row operations may change the linear dependence relations among the rows of a matrix.

Hence, the given statement is false.

Step 3: Use the rank theorem

(c)

The sum of the pivot columns and non-pivot columns of A equals the number of columns of A. The number of pivot columns of A is the rank of A.

By the rank theorem, the number of non-pivot columns of A is the dimension of the null space of A.

Hence, the given statement is true.

Step 4: Use the fact of the transpose matrix

(d)

The rows of \({A^T}\) are identified with the columns of \({\left( {{A^T}} \right)^T}\), i.e., A. So, you can write \({\rm{Row }}{A^T}\) in place of Col A.

Hence, the given statement is true.

Step 5: Use the 1st statement of theorem 13

(e)

By Theorem 13, if matrices A and B are equivalent, then their row spaces are the same.

Hence, the given statement is true.

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Linear Algebra and its Applications

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Short Answer

Step by Step Solution

Step 1: Use the fact of pivot columns

Step 2: Use the properties of a basis

Step 3: Use the rank theorem

Step 4: Use the fact of the transpose matrix

Step 5: Use the 1st statement of theorem 13

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Linear Algebra and its Applications

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Short Answer

Step by Step Solution

Step 1: Use the fact of pivot columns

Step 2: Use the properties of a basis

Step 3: Use the rank theorem

Step 4: Use the fact of the transpose matrix

Step 5: Use the 1st statement of theorem 13

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