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.
