Math Problem Statement
For p \in P(\mathbb{R}), define Tp:\mathbb{R} \rightarrow \mathbb{R} by (Tp)(x) = \begin{cases} \frac{p(x) - p(3)}{x-3} & if x \neq 3 \ p'(3) & if x = 3 \end{cases} for each x \in \mathbb{R}. Show that Tp \in P(\mathbb{R}) for every polynomial p \in P(\mathbb{R}) and show that T:P(\mathbb{R}) \rightarrow P(\mathbb{R}) is a linear map.
Solution
To solve this problem, we need to demonstrate two key properties:
- That for every polynomial : This means , as defined, is itself a polynomial.
- That is a linear map: This requires showing that satisfies linearity.
Step 1:
Case 1: When
The definition of for is: For , is a polynomial of degree . Subtracting , a constant, does not change the degree of the polynomial. Dividing by reduces the degree by 1. Hence, is a polynomial of degree .
Case 2: When
For , . The derivative of a polynomial is itself a polynomial (of degree ). Therefore, is well-defined as a specific value of this polynomial.
Combining the two cases
For all , is defined piecewise. At , matches the value of the polynomial for due to the continuity of polynomials and derivatives. Hence, is a polynomial of degree .
Thus, .
Step 2: Linearity of
To show is a linear map, we need to verify two properties:
- for all ,
- for all and .
Property 1:
For , Expanding the numerator: Thus, At , using the definition: Thus, for all .
Property 2:
For , Since , At , using the definition: Thus, for all .
Conclusion
- for every polynomial .
- is a linear map.
Let me know if you'd like further clarifications or details on any part of the proof! Here are five follow-up questions:
- Why is the continuity of important at ?
- How does the degree reduction in affect the image of under ?
- Can be represented in terms of a matrix if is finite-dimensional? How?
- What happens if we apply multiple times to the same polynomial ?
- How does behave when applied to a constant polynomial?
Tip: Always verify that piecewise-defined functions are continuous to ensure smooth transitions across cases!
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Math Problem Analysis
Mathematical Concepts
Polynomial functions
Linear maps
Piecewise functions
Continuity and differentiability
Formulas
Tp(x) = \frac{p(x) - p(3)}{x - 3} \text{ for } x \neq 3
Tp(3) = p'(3)
Theorems
Linearity of differentiation
Degree reduction of polynomials
Suitable Grade Level
Undergraduate level
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