Suggested languages for you:

Americas

Europe

Q18E

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 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.

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

## 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.