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p(0 Define T : P2 → R² by T(p) = [p(1). 3 For example if p(t) = 3+ 4t + 5t² then T(p) = 12 (a) Show that T is a linear transformation. (b) Find a polynomial in the kernel of T. (c) What is the range of T?

p(0 Define T : P2 → R² by T(p) = [p(1). 3 For example if p(t) = 3+ 4t + 5t² then T(p) = 12 (a) Show that T is a linear transformation. (b) Find a polynomial in the kernel of T. (c) What is the range of T?

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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4. Define \( T : \mathbb{P}_2 \rightarrow \mathbb{R}^2 \) by \( T(p) = \begin{bmatrix} p(0) \\ p(1) \end{bmatrix} \). For example, if \( p(t) = 3 + 4t + 5t^2 \), then \( T(p) = \begin{bmatrix} 3 \\ 12 \end{bmatrix} \). (a) Show that \( T \) is a linear transformation. (b) Find a polynomial in the kernel of \( T \). (c) What is the range of \( T \)?
Transcribed Image Text:4. Define \( T : \mathbb{P}_2 \rightarrow \mathbb{R}^2 \) by \( T(p) = \begin{bmatrix} p(0) \\ p(1) \end{bmatrix} \). For example, if \( p(t) = 3 + 4t + 5t^2 \), then \( T(p) = \begin{bmatrix} 3 \\ 12 \end{bmatrix} \). (a) Show that \( T \) is a linear transformation. (b) Find a polynomial in the kernel of \( T \). (c) What is the range of \( T \)?
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