Math 152 Sec4.6

If B is any echelon form of​ A, then the pivot columns of B form a basis for the column space of A. Is this statement true or​ false?

The statement is false. The columns of an echelon form B of A are often not in the column space of A.

The row space of AT is the same as the column space of A. Is this statement true or​ false?

The statement is true because the rows of AT are the columns of left parenthesis Upper A Superscript Upper T right parenthesis Superscript Upper T Baseline equals Upper A.

Row operations preserve the linear dependence relations among the rows of A. Is this statement true or​ false?

The statement is false. Row operations may change the linear dependence relations among the rows of A.

If A and B are row​ equivalent, then their row spaces are the same. Is this statement true or​ false?

The statement is true. If B is obtained from A by row​ operations, the rows of B are linear combinations of the rows of A and​ vice-versa

The dimension of the null space of A is the number of columns of A that are not pivot columns. Is this statement true or​ false?

The statement is true. The dimension of Nul A equals the number of free variables in the equation Ax=0.