Are the following statements true or false? If the system Ax = b is inconsistent

Question

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

Explanation / Answer

1.True.

2.True.

4.True.

If the columns span R^m,this says that every b inR^m is in the span of columns i.e., b is a linear combination of columns.Then the system is consistent.

5.False.

The system is inconsistent if the augmented matrix has pivot in the last("b") column.

6.False

7.True.

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