Wronskian vs. Determinant in Determining Linear Independence?

ManessIn summary, the Wronskian is a type of determinant used to determine the linear independence of solutions to a differential equation. It is specifically used for function space, while determinants are used for vectors in cartesian space. The Wronskian is necessary in this case to determine if a row in a matrix can be zeroed out through Gaussian reduction, and therefore if one of the equations or vectors that constructed the matrix is dependent on the other two basis vectors.
  • #1
kq6up
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It seems to me that if a row is able to be zeroed out through Gaussian reduction that the determinate of that matrix would equal zero. Therefore, we know that at least one of equations/vectors that constructed the matrix was formed from the other two rows. That is -- that equation is dependent on the other two basis vectors.

Why do we need the Wronskian to determine this?

Thanks,
Chris Maness
 
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  • #2
The Wronskian is a special type of determinant used to determine if a set of solutions to a differential equation is linearly independent:

http://en.wikipedia.org/wiki/Wronskian

See the section on "The Wronskian and linear independence".
 
  • #3
Ok, is it that Wronskians are for function space where all the basis are formed by functions of x, where the determinants are for -- say -- {x,y,z} vectors in cartesian space?

Chris
 

1. What is the difference between Wronskian and Determinant?

The Wronskian is a special type of determinant that is used to determine the linear independence of a set of functions. The determinant, on the other hand, is a mathematical tool used to solve systems of linear equations.

2. How are Wronskian and Determinant calculated?

The Wronskian is calculated by taking the determinant of a matrix composed of the functions and their derivatives. The determinant is calculated by using a specific formula based on the size of the matrix.

3. Can Wronskian and Determinant be used interchangeably to determine linear independence?

No, Wronskian and Determinant serve different purposes and cannot be used interchangeably. The Wronskian is specifically used to test the linear independence of a set of functions, while the determinant is used to solve systems of linear equations.

4. How do Wronskian and Determinant help in determining linear independence?

If the Wronskian of a set of functions is non-zero at a specific point, then the functions are linearly independent at that point. Similarly, if the determinant of a matrix representing a set of equations is non-zero, then the system of equations has a unique solution and the equations are linearly independent.

5. Are there any other factors to consider besides the Wronskian and Determinant when determining linear independence?

Yes, there are other methods and criteria that can be used to determine linear independence, such as the rank of a matrix, the span of a set of vectors, and the use of linear transformations.

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