8. Let T₁ R2 → P₁ (R) and T₂: P₁ (R)→ R2 be the linear transformations given by: (a₁ + a²) + (a₁ − a₂)x, T₂(a+bx) = [(a+b)/²] Ti Find T₁T2 and T₂T₁.
8. Let T₁ R2 → P₁ (R) and T₂: P₁ (R)→ R2 be the linear transformations given by: (a₁ + a²) + (a₁ − a₂)x, T₂(a+bx) = [(a+b)/²] Ti Find T₁T2 and T₂T₁.
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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Question
![8. Let \( T_1 : \mathbb{R}^2 \rightarrow P_1(\mathbb{R}) \) and \( T_2 : P_1(\mathbb{R}) \rightarrow \mathbb{R}^2 \) be the linear transformations given by:
\[ T_1\left( \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \right) = (a_1 + a_2) + (a_1 - a_2)x, \quad T_2(a + bx) = \begin{bmatrix} (a + b)/2 \\ (a - b)/2 \end{bmatrix}. \]
Find \( T_1T_2 \) and \( T_2T_1 \).
9. Let \( T : M_2(\mathbb{R}) \rightarrow M_2(\mathbb{R}) \) map a \( 2 \times 2 \) matrix to its transpose, i.e., \( T(A) = A^T \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67429e19-8d15-47bc-bbfe-dfa923849540%2F34c1009e-3b65-4cc9-ad60-c60c7d248a77%2Ftnd38to_processed.jpeg&w=3840&q=75)
Transcribed Image Text:8. Let \( T_1 : \mathbb{R}^2 \rightarrow P_1(\mathbb{R}) \) and \( T_2 : P_1(\mathbb{R}) \rightarrow \mathbb{R}^2 \) be the linear transformations given by:
\[ T_1\left( \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \right) = (a_1 + a_2) + (a_1 - a_2)x, \quad T_2(a + bx) = \begin{bmatrix} (a + b)/2 \\ (a - b)/2 \end{bmatrix}. \]
Find \( T_1T_2 \) and \( T_2T_1 \).
9. Let \( T : M_2(\mathbb{R}) \rightarrow M_2(\mathbb{R}) \) map a \( 2 \times 2 \) matrix to its transpose, i.e., \( T(A) = A^T \).
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