Find a linear differentiation transformation

In summary, the conversation discusses a linear transformation T that maps functions with first and second derivatives to their second derivative functions. The question is raised about the kernel of this transformation, which is defined as the set of functions whose second derivatives are zero. It is noted that one direction of this is easy to prove, while the other direction requires more delicate reasoning. However, there are potential issues with the map and the question may need to be rephrased to avoid these issues.
  • #1
yanyin
21
0
let C^2 be the set of functions with domain R with have first and second derivatives at all points. Defind a linear transformation T: C^2 > C^2 by T(f) = f'', in other words, each input function is mapped to its second derivative function.
what is the kernel of this transformation?
 
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  • #2
that isn't the usual definition of C^2, and moreover the map you define isn't a map from C^2 to C^2, but apart from that...
 
  • #3
the kernel is the set of vectors, in this case functions, that are mapped to the zero vector, in this case the zero function.

so your question is the same as asking you to list all functions whose second derivatives are zero. one direction is pretty easy: it's a certain type of polynomial. the other direction, proving that those are the only ones with vanishing second derivatives, is a bit harder and the easiest way i can think of to prove that is to apply the mean value theorem twice (or rolle's theorem).
 
  • #4
it genuinely is more delicate than that - the map given is not well defined. Let f be a function that does not have a third derivative, but has the first two, then T(f) is not in C^2 (there ought to be a constraint on the second derivative being continuous too) T is a map from C^{n+2} to C^{n}. Example: integrate |x| twice the resulting function is twice continuously differentiable, but its image under T is not in C^2, hell it's not even in C^1
 
  • #5
hmm... i guess i overestimated the power of underestimation.
 
  • #6
if the question were rephrased to sidestep these issues then what you did is valid
 

What is a linear differentiation transformation?

A linear differentiation transformation is a mathematical operation that transforms a set of data points into a straight line, while preserving the relative distances between the points. It is commonly used in statistics and data analysis to simplify data and make it easier to interpret.

How is a linear differentiation transformation different from other transformations?

A linear differentiation transformation is unique because it preserves the original shape and relative distances of the data points. Other transformations, such as logarithmic or exponential transformations, change the shape of the data and the distances between points.

What is the purpose of using a linear differentiation transformation?

The purpose of a linear differentiation transformation is to make data more manageable and easier to analyze. By transforming data into a straight line, it becomes easier to identify patterns and trends, and to make predictions based on the data.

How do you find a linear differentiation transformation?

To find a linear differentiation transformation, you need to perform a series of mathematical calculations on the data. This typically involves calculating the slope and intercept of the line that best fits the data points, using techniques such as least squares regression.

Are there any limitations to using a linear differentiation transformation?

Yes, there are limitations to using a linear differentiation transformation. It is only applicable to data that can be represented as a straight line, and it may not accurately represent the true relationship between variables in some cases. Additionally, it may not be appropriate for data sets with extreme outliers or non-linear relationships.

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