Is A Full Rank Equivalent to an Overdetermined System?

AI Thread Summary
A full rank matrix does not necessarily imply that the system is overdetermined. An overdetermined system, characterized by more equations than unknowns, can have a full rank matrix but may not always have an exact solution. The statement from the book highlights that a full column rank matrix often leads to no exact solution for Ax=b. Therefore, while full rank is a common feature of overdetermined systems, they are not equivalent. Understanding this distinction is crucial for solving such systems effectively.
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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