Question

In Exercise 17, Ais 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 Ais 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.

Short answer

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.

Step-by-step solution

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.

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.

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.

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.