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Q17E

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 17, A is an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.**

**17. a. The row space of A is the same as the column space of \({A^T}\].**

** b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.**

** c. The dimensions of the row space and the column space of A are the same, even if A is not square.**

** d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.**

** e. On a computer, row operations can change the apparent rank of a matrix.**

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

(a)

Since the rows of *A* are identified with the columns of \({A^T}\], you can write \({\rm{Col }}{A^T}\] in place of Row *A*.

Hence, the given statement is true.

(b)

Although the first three rows of *B* are **linearly independent**, it is wrong to conclude that the first three rows of *A* are linearly independent. The **row operations may change the linear dependence relations among the rows of a matrix**.

Hence, the given statement is false.

(c)

By **the rank theorem**, the dimensions of the column space and the row space of an \(m \times n\] matrix *A* are equal.

Hence, the given statement is true.

(d)

By the rank theorem, the sum of the dimensions of the row space and the null space of *A* equals the number of pivot positions in *A*. The number of pivot positions need not be equal to the number of rows of *A*.

Hence, the given statement is false.

(e)

On a computer, the exact arithmetic is performed on a matrix whose entries are specified, but the row operations can change the apparent rank of a matrix.

Hence, the given statement is true.

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