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Q18E

Expert-verifiedFound in: Page 191

Book edition
5th

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

Pages
483 pages

ISBN
978-03219822384

**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.

(a)

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

Hence, the given statement is false.

(b)

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

Hence, the given statement is false.

(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.

(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.

(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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