Solving the System Ax = b: Is Full Rank Necessary?

In summary, the statement is false because b does not need to be in the range of A for the system Ax=b to have a solution. A counterexample would be a simple A matrix with columns that are elements of the standard basis for \mathbb{R}^m.
  • #1
BobSea
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Homework Statement


The system Ax=b, with Amxn, and m>n, always has a solution when A has full rank. If False, give a counter example, if True, say why.



Homework Equations


None


The Attempt at a Solution


I want to say False because b doesn't need to be in the range of A, so Ax=b wouldn't have a solution. However, I'm having trouble making a counter example (i.e. A = 3x2 matrix and b = 3x1 matrix) that proves the point.
 
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  • #2
Your reasoning is correct. For a counterexample, try a very simple A whose columns are elements of the standard basis for [itex]\mathbb{R}^m[/itex].
 

FAQ: Solving the System Ax = b: Is Full Rank Necessary?

What does it mean for a system to have "full rank"?

Having full rank means that the system of equations has enough independent equations to uniquely determine the solutions for all variables in the system. In other words, there are no redundant or dependent equations in the system.

Why is full rank necessary for solving a system of linear equations?

Full rank is necessary because it ensures that the system has a unique solution. Without full rank, there may be multiple solutions or no solutions at all. This can lead to inconsistencies and inaccuracies in solving the system.

Can a system with less than full rank still be solved?

Yes, a system with less than full rank can still be solved, but it may not have a unique solution. In this case, it may have either no solutions or an infinite number of solutions. It is important to note that the solutions obtained from a system with less than full rank may not be accurate or meaningful.

How can I determine if a system has full rank?

To determine if a system has full rank, you can use the rank-nullity theorem, which states that the rank of a matrix is equal to the number of non-zero rows in its reduced row echelon form. If the rank of the matrix is equal to the number of variables in the system, then the system has full rank.

What happens if a system does not have full rank?

If a system does not have full rank, it means that the system is either inconsistent (has no solutions) or has infinitely many solutions. In this case, the system cannot be solved using traditional methods and may require further analysis and techniques such as least squares approximation.

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