Math Problem Statement

a) T1 : P(F) → P(F) p(x) 7→ xp(x) + p ′ (x)

Solution

Let's analyze the transformation T1:P(F)P(F)T_1: P(F) \rightarrow P(F) given by T1(p(x))=xp(x)+p(x)T_1(p(x)) = x p(x) + p'(x)

where:

  • p(x)p(x) is a polynomial in P(F)P(F),
  • p(x)p'(x) represents the derivative of p(x)p(x) with respect to xx,
  • FF is the field over which the polynomial p(x)p(x) is defined.

Part (a): Determine the Properties of T1T_1

To understand the properties of T1T_1, 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 T1T_1 is a linear transformation, we need to check if for any polynomials p(x),q(x)P(F)p(x), q(x) \in P(F) and scalars a,bFa, b \in F, the following holds: T1(ap(x)+bq(x))=aT1(p(x))+bT1(q(x))T_1(a p(x) + b q(x)) = a T_1(p(x)) + b T_1(q(x))

Calculation:
  1. Consider T1(ap(x)+bq(x))T_1(a p(x) + b q(x)): T1(ap(x)+bq(x))=x(ap(x)+bq(x))+(ap(x)+bq(x))T_1(a p(x) + b q(x)) = x(a p(x) + b q(x)) + (a p(x) + b q(x))'
  2. Expanding each term: =axp(x)+bxq(x)+ap(x)+bq(x)= a x p(x) + b x q(x) + a p'(x) + b q'(x)
  3. Grouping terms, we get: =a(xp(x)+p(x))+b(xq(x)+q(x))= a (x p(x) + p'(x)) + b (x q(x) + q'(x))
  4. This simplifies to: =aT1(p(x))+bT1(q(x))= a T_1(p(x)) + b T_1(q(x))

Since the expression holds, T1T_1 is linear.

2. Transformation of Specific Polynomials:

Let's evaluate T1T_1 on some basic polynomials to understand its action better.

  1. Constant Polynomial p(x)=cp(x) = c:

    • Here, p(x)=cp(x) = c and p(x)=0p'(x) = 0.
    • Then: T1(c)=xc+0=cxT_1(c) = x \cdot c + 0 = c \cdot x
  2. Linear Polynomial p(x)=xp(x) = x:

    • For p(x)=xp(x) = x, p(x)=1p'(x) = 1.
    • Then: T1(x)=xx+1=x2+1T_1(x) = x \cdot x + 1 = x^2 + 1
  3. Quadratic Polynomial p(x)=x2p(x) = x^2:

    • For p(x)=x2p(x) = x^2, p(x)=2xp'(x) = 2x.
    • Then: T1(x2)=xx2+2x=x3+2xT_1(x^2) = x \cdot x^2 + 2x = x^3 + 2x

These examples show that T1T_1 transforms polynomials by increasing the degree of each polynomial by one, adding both a term involving xx times the polynomial and the derivative of the polynomial.

Would you like further details on the properties of this transformation?

Related Questions

  1. What is the kernel of T1T_1 for polynomials of degree 1?
  2. How does T1T_1 affect higher-degree polynomials?
  3. Is T1T_1 an invertible transformation?
  4. What is the range of T1T_1 on polynomials of a fixed degree?
  5. Can T1T_1 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