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GIVEN: P₂ = { a₁ + a₁x + a₂x² ao, a₁, a₂ ≤ R} T: P₂ R, T(a₁ + a₁x + a₂x²) = a₂ For example: 7(1- x + 3x²) = 3 PROVE: T is a linear transformation. -

GIVEN: P₂ = { a₁ + a₁x + a₂x² ao, a₁, a₂ ≤ R} T: P₂ R, T(a₁ + a₁x + a₂x²) = a₂ For example: 7(1- x + 3x²) = 3 PROVE: T is a linear transformation. -

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Given:** \( P_2 = \{ a_0 + a_1 x + a_2 x^2 \mid a_0, a_1, a_2 \in \mathbb{R} \} \) **Transformation:** \( T: P_2 \rightarrow \mathbb{R} \), where \( T(a_0 + a_1 x + a_2 x^2) = a_2 \). *For example:* \( T(1 - x + 3x^2) = 3 \). **Prove:** \( T \) is a linear transformation. --- In the above content, \( P_2 \) is the space of all polynomials of degree at most 2 with real coefficients. The transformation \( T \) maps a quadratic polynomial to the coefficient of \( x^2 \). To prove that \( T \) is a linear transformation, we need to show that it satisfies the properties of linearity: 1. **Additivity:** \( T(f + g) = T(f) + T(g) \) for any polynomials \( f, g \in P_2 \). 2. **Homogeneity:** \( T(c \cdot f) = c \cdot T(f) \) for any polynomial \( f \in P_2 \) and scalar \( c \in \mathbb{R} \).
Transcribed Image Text:**Given:** \( P_2 = \{ a_0 + a_1 x + a_2 x^2 \mid a_0, a_1, a_2 \in \mathbb{R} \} \) **Transformation:** \( T: P_2 \rightarrow \mathbb{R} \), where \( T(a_0 + a_1 x + a_2 x^2) = a_2 \). *For example:* \( T(1 - x + 3x^2) = 3 \). **Prove:** \( T \) is a linear transformation. --- In the above content, \( P_2 \) is the space of all polynomials of degree at most 2 with real coefficients. The transformation \( T \) maps a quadratic polynomial to the coefficient of \( x^2 \). To prove that \( T \) is a linear transformation, we need to show that it satisfies the properties of linearity: 1. **Additivity:** \( T(f + g) = T(f) + T(g) \) for any polynomials \( f, g \in P_2 \). 2. **Homogeneity:** \( T(c \cdot f) = c \cdot T(f) \) for any polynomial \( f \in P_2 \) and scalar \( c \in \mathbb{R} \).
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