-1 98 dit 10. (a) Given the following mapping T R³ → R2. Determine whether T is a linear : transformation or not. T(x1, x2, 3) = (x₁ + 3x2 + x3, x1 - x2) (b) Given the following mapping T: P₂ (R) → R. Determine whether T is a linear transformation or not. O T(a+bx+cx²) = a +b+c+1
-1 98 dit 10. (a) Given the following mapping T R³ → R2. Determine whether T is a linear : transformation or not. T(x1, x2, 3) = (x₁ + 3x2 + x3, x1 - x2) (b) Given the following mapping T: P₂ (R) → R. Determine whether T is a linear transformation or not. O T(a+bx+cx²) = a +b+c+1
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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![Certainly! Here's a transcription suitable for an educational website:
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**Problem 10**
**(a)** Given the following mapping \( T : \mathbb{R}^3 \to \mathbb{R}^2 \). Determine whether \( T \) is a linear transformation or not.
\[
T(x_1, x_2, x_3) = (x_1 + 3x_2 + x_3, x_1 - x_2)
\]
**(b)** Given the following mapping \( T : P_2(\mathbb{R}) \to \mathbb{R} \). Determine whether \( T \) is a linear transformation or not.
\[
T(a + bx + cx^2) = a + b + c + 1
\]
---
This task involves checking whether the given mappings satisfy conditions for linearity, specifically additivity and homogeneity.
- For part (a), you assess if the function \( T(x_1, x_2, x_3) \) maintains these properties across vectors in \( \mathbb{R}^3 \).
- For part (b), evaluate if the transformation of quadratic polynomials to real numbers meets the linearity conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67429e19-8d15-47bc-bbfe-dfa923849540%2F83e09934-8902-4b78-92bc-1eecaf41445e%2Fbhf6bwj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's a transcription suitable for an educational website:
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**Problem 10**
**(a)** Given the following mapping \( T : \mathbb{R}^3 \to \mathbb{R}^2 \). Determine whether \( T \) is a linear transformation or not.
\[
T(x_1, x_2, x_3) = (x_1 + 3x_2 + x_3, x_1 - x_2)
\]
**(b)** Given the following mapping \( T : P_2(\mathbb{R}) \to \mathbb{R} \). Determine whether \( T \) is a linear transformation or not.
\[
T(a + bx + cx^2) = a + b + c + 1
\]
---
This task involves checking whether the given mappings satisfy conditions for linearity, specifically additivity and homogeneity.
- For part (a), you assess if the function \( T(x_1, x_2, x_3) \) maintains these properties across vectors in \( \mathbb{R}^3 \).
- For part (b), evaluate if the transformation of quadratic polynomials to real numbers meets the linearity conditions.
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