eyehategod said:also it would mean that b is in the column space of A.
eyehategod said:i guess my problem is that i don't quite understand when it says "consistent for all column vectors b."
A matrix equation, Ax= b, is "consistent" if it has at least one solution. "Ax= b is consistent for all b" means the equation Ax= b is consistent no matter what vector b is.radou said:A system of linear equations is consistent if it has a solution. Of course, this solution need not be unique.
A linear system is consistent if it has at least one solution. This means that there is a set of values for the variables that satisfies all of the equations in the system.
The rank of a matrix is the number of linearly independent rows or columns in the matrix. In other words, it is the number of equations or variables in the system that are not redundant or dependent on each other.
To prove that a linear system Ax=b is consistent, you need to show that the rank of the coefficient matrix A is equal to the number of variables in the system. This can be done through various methods such as row reduction or the use of the Rank-Nullity Theorem.
A linear system is inconsistent if it has no solution. This means that there is no set of values for the variables that satisfies all of the equations in the system. In other words, the equations in the system are contradictory and cannot be satisfied simultaneously.
Proving the consistency of a linear system is important because it allows us to determine whether the system has solutions or not. It also helps us to understand the relationships between the variables and equations in the system and can lead to further insights and applications in various fields of science and engineering.