Wronskian vs. Determinant in Determining Linear Independence?

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The discussion centers on the relationship between the Wronskian and the determinant in determining linear independence. It is noted that if a row in a matrix can be zeroed out through Gaussian reduction, the determinant becomes zero, indicating dependence among the vectors. The Wronskian is highlighted as a specific type of determinant useful for assessing linear independence in the context of differential equations. There is a distinction made between the use of Wronskians for function spaces and determinants for vector spaces like Cartesian coordinates. Understanding these concepts is crucial for correctly identifying linear independence in different mathematical contexts.
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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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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".
 
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
 

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