The Meaning of Degenerate in the Context of Linear Algebra

In summary, degeneracy in this context refers to the derivative map having a non-trivial null space. This means that the function and its derivative have a relationship where the derivative is equal to zero at certain points. This property holds for functions in the set \(R[x]_{n}\) but not for the set \(<\sin t,\,\cos t>\).
  • #1
Sudharaka
Gold Member
MHB
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Hi everyone, :)

Here's a question I encountered.

Show that \(f(x)\rightarrow f'(x)\) is degenerate on \(R[x]_{n}\) and is non-degenerate on \(<\sin t,\,\cos t>\)

Don't give me the full answer but explain what is meant by degeneracy in this context.

Thank you.
 
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  • #2
It means the derivative map has a non-trivial null space.
 
  • #3
Deveno said:
It means the derivative map has a non-trivial null space.

Thank you very much. Now I understand it perfectly. I am doing an Advanced Linear Algebra course and I am not getting everything perfectly. I think I have to put a lot of effort into it. :)
 

FAQ: The Meaning of Degenerate in the Context of Linear Algebra

What does degenerate mean in the context of linear algebra?

In linear algebra, degenerate refers to a special case where a system of equations has infinitely many solutions or no unique solution at all. This can occur when there is redundant information or when the equations are inconsistent.

How do you identify a degenerate system in linear algebra?

A degenerate system can be identified by examining the coefficient matrix of the system. If the determinant of the matrix is equal to zero, the system is degenerate. Additionally, if the number of equations is less than the number of variables, the system is also degenerate.

What are some real-world applications of degenerate systems in linear algebra?

One common application of degenerate systems is in solving optimization problems, where the goal is to find the maximum or minimum value of a function. Degenerate systems can also arise in economics, physics, and engineering when trying to find equilibrium points or solutions to complex systems.

How do degenerate systems affect the solution process in linear algebra?

In a degenerate system, the usual methods of solving linear equations may not work, as there is no unique solution. Instead, special techniques such as Gaussian elimination with partial pivoting or using a generalized inverse may be necessary to find a solution.

Can degenerate systems have any solutions in linear algebra?

Yes, degenerate systems can have infinitely many solutions or no solutions at all, depending on the specific system. In some cases, degeneracy can lead to unexpected or unusual results, so it is important to identify and handle these systems carefully in order to accurately solve problems in linear algebra.

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