Is A Full Rank Equivalent to an Overdetermined System?

In summary, a full rank system has an equal number of independent equations and unknown variables, resulting in a unique solution. It is different from an overdetermined system, which has more equations than unknown variables and may have no solution or an infinite number of solutions. A full rank system can also be underdetermined, with fewer equations than unknown variables and may have an infinite number of solutions. The rank of a system can be determined through methods such as Gaussian elimination or by calculating the determinant of the coefficient matrix. A full rank system cannot become overdetermined, but adding additional equations may make it inconsistent with no solutions.
  • #1
Niles
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Homework Statement


Hi

If I am dealing with an overdetermined system Ax=b, then I can (assuming A has full rank) find the unique approximative solution by least squares.

Now, in my book it says that: "For a full column rank matrix, it is frequently the case that no solution x satisfies Ax=b exactly". I assume the book is saying that A having full rank is equivalent to it being overdetermined.

Is that always the case?


Niles.
 
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  • #2
That is not always the case, I found out.
 

FAQ: Is A Full Rank Equivalent to an Overdetermined System?

What does it mean for a system to be full rank?

A full rank system is one in which the number of independent equations is equal to the number of unknown variables. This means that the system has a unique solution and is not underdetermined or overdetermined.

How is a full rank system different from an overdetermined system?

A full rank system has exactly enough equations to determine a unique solution, while an overdetermined system has more equations than unknown variables. This means that an overdetermined system may have no solution or an infinite number of solutions.

Is a full rank system always overdetermined?

No, a full rank system can also be underdetermined if there are fewer equations than unknown variables. In this case, the system may have an infinite number of solutions.

How do you determine if a system is full rank or overdetermined?

A system is full rank if its rank is equal to the number of variables. The rank of a system can be determined through methods such as Gaussian elimination or by calculating the determinant of the coefficient matrix.

Can a full rank system become overdetermined?

No, a full rank system cannot become overdetermined. However, if additional equations are added to a full rank system, it may become inconsistent and have no solutions.

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