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.

(a)

Note that thepivot 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.

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.

(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.