Let T: P2 P3 be the linear transformation defined by .3x T(p(x)) = (x + 1)p(x) + p′(1) + Find [7], the matrix of T relative to the standard bases of P₂ ({1, x, x²}) and P3 ({1, x, x², x³ }). 2p(t) dt.

Let T: P2 P3 be the linear transformation defined by .3x T(p(x)) = (x + 1)p(x) + p′(1) + Find [7], the matrix of T relative to the standard bases of P₂ ({1, x, x²}) and P3 ({1, x, x², x³ }). 2p(t) dt.

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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I'm currently facing challenges in solving this problem using only matrix notation, and I'm hoping you can help me. The problem specifically requires a solution using matrix notation exclusively, without any other methods. Could you please provide a detailed explanation, step by step, using matrix notation, to guide me towards the final solution?

### Linear Transformation and Matrix Representation

Let $ T : P_2 \rightarrow P_3 $ be the linear transformation defined by

\[
T \left( p(x) \right) = (x + 1)p(x) + p'(1) + \int_0^{3x} 2p(t) \, dt.
\]

Find $[T]$, the matrix of $T$ relative to the standard bases of $ P_2 \left(\{1, x, x^2\}\right) $ and $ P_3 \left(\{1, x, x^2, x^3\}\right) $.

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