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Expert-verified Found in: Page 191 ### Linear Algebra and its Applications

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.

1. The statement is true.
2. The statement is false.
3. The statement is true.
4. The statement is false.
5. The statement is true.
See the step by step solution

## Step 1: Use the fact of the transpose matrix

(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.

## Step 2: Use the properties of a basis

(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.

## Step 3: Use the rank theorem

(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.

## Step 4: Use the numerical note before the practice problem

(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. ### Want to see more solutions like these? 