Question

Are the following statements true or false? 1. If the augmented matrix [A | b] has a pivot position in every row, then the system Ax = b is inconsistent. 2. If A is an m x n matrix and if the equation Ax = b is inconsistent for some b in R^m, then the RREF of A cannot have a pivot position in every row. 3. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x. 4. If the columns of an m x n matrix A span R^m, then the equation Ax = b is consistent for each b in R^m 5. If the system Ax = b is inconsistent, then b is not in the column space of A. 6. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution. 7. The equation Ax = b is referred to as a vector equation.

          Are the following statements true or false?
1. If the augmented matrix [A | b] has a pivot position in every row, then the system Ax = b is inconsistent.
2. If A is an m x n matrix and if the equation Ax = b is inconsistent for some b in R^m, then the RREF of A cannot have a pivot position in every row.
3. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.
4. If the columns of an m x n matrix A span R^m, then the equation Ax = b is consistent for each b in R^m
5. If the system Ax = b is inconsistent, then b is not in the column space of A.
6. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution.
7. The equation Ax = b is referred to as a vector equation.

Are the following statements true or false?
1. If the augmented matrix [A | b] has a pivot position in every row, then the system Ax = b is inconsistent.
2. If A is an m x n matrix and if the equation Ax = b is inconsistent for some b in R^m, then the RREF of A cannot have a pivot position in every row.
3. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.
4. If the columns of an m x n matrix A span R^m, then the equation Ax = b is consistent for each b in R^m
5. If the system Ax = b is inconsistent, then b is not in the column space of A.
6. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution.
7. The equation Ax = b is referred to as a vector equation.

Added by Olga S.

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