Math Problem Statement
a) T1 : P(F) → P(F) p(x) 7→ xp(x) + p ′ (x)
Solution
Let's analyze the transformation given by
where:
- is a polynomial in ,
- represents the derivative of with respect to ,
- is the field over which the polynomial is defined.
Part (a): Determine the Properties of
To understand the properties of , we need to verify if it has certain characteristics, such as linearity, and identify its action on specific polynomials.
1. Linearity Check:
To confirm if is a linear transformation, we need to check if for any polynomials and scalars , the following holds:
Calculation:
- Consider :
- Expanding each term:
- Grouping terms, we get:
- This simplifies to:
Since the expression holds, is linear.
2. Transformation of Specific Polynomials:
Let's evaluate on some basic polynomials to understand its action better.
-
Constant Polynomial :
- Here, and .
- Then:
-
Linear Polynomial :
- For , .
- Then:
-
Quadratic Polynomial :
- For , .
- Then:
These examples show that transforms polynomials by increasing the degree of each polynomial by one, adding both a term involving times the polynomial and the derivative of the polynomial.
Would you like further details on the properties of this transformation?
Related Questions
- What is the kernel of for polynomials of degree 1?
- How does affect higher-degree polynomials?
- Is an invertible transformation?
- What is the range of on polynomials of a fixed degree?
- Can be expressed in matrix form for a specific basis?
Tip
To verify linear transformations, always check both linearity properties: additivity and scalar multiplication, as they confirm if the transformation maintains polynomial structure.
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Math Problem Analysis
Mathematical Concepts
Linear Transformation
Polynomials
Derivative
Field Theory
Formulas
T1(p(x)) = x p(x) + p'(x)
Theorems
Linearity Test for Transformations
Suitable Grade Level
College/Undergraduate
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