Question

In Exercise 18, Ais 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.

Short answer

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.

Step-by-step solution

Use the fact of pivot columns

(a)

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

Hence, the given statement is false.

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.

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.

Bythe 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.

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.

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.