8 second by using the matrix you found in part (a)8) Define a linear T: R2[x] → R3[x] by T(p(x)) = x²p"(x) – 2p'(x) + xp(x), wherep'(x) and p"(x) are the first and second derivatives of the polynomial p(x), respectively.Determine the matrix of T relative to the standard basis of R2[x] and Rg(x].9) Suppose V is a vector space and S,T E L(V,V) are such that Range S c ker T =null T.

8 Define a linear map T:R2x→R3x by Tpx=x2p''x−2p'x+xpx. We have to determine the matrix T relative…

Answered: 8) Define a linear T: R2[x] → R3[x] by…

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

second by using the matrix you found in part (a)
8) Define a linear T: R2[x] → R3[x] by T(p(x)) = x²p"(x) – 2p'(x) + xp(x), where
p'(x) and p"(x) are the first and second derivatives of the polynomial p(x), respectively.
Determine the matrix of T relative to the standard basis of R2[x] and Rg(x].
9) Suppose V is a vector space and S,T E L(V,V) are such that Range S c ker T =null T.

Expert Solution

Step 1

$(8)$ Define a linear map $T : R_{2} [x] \to R_{3} [x]$ by $T (p (x)) = x^{2} p^{''} (x) - 2 p^{'} (x) + x p (x)$ .

We have to determine the matrix $T$ relative to the standard basis of $R_{2} [x]$ and $R_{3} [x]$ .

Let the basis for $R_{2} [x]$ is $\{1, x, x^{2}\}$ and basis of $R_{3} [x]$ is $\{1, x, x^{2}, x^{3}\}$ .

$\begin{matrix} T (1) = x^{2} \cdot 0 - 2 \cdot 0 + x \cdot 1 (Sin c e, p^{''} (x) = 0 a n d p^{'} (x) = 0) \\ = x \\ = 0 \cdot 1 + 1 \cdot x + 0 \cdot x^{2} + 0 \cdot x^{3} \end{matrix}$

Hence, $T (1) = x = 0 \cdot 1 + 1 \cdot x + 0 \cdot x^{2} + 0 \cdot x^{3}$

Now,

$\begin{matrix} T (x) = x^{2} \cdot 0 - 2 \cdot 1 + x \cdot x (Sin c e, p^{''} (x) = 0 a n d p^{'} (x) = 1) \\ = - 2 + x^{2} \\ = - 2 \cdot 1 + 0 \cdot x + 1 \cdot x^{2} + 0 \cdot x^{3} \end{matrix}$

Hence, $T (x) = - 2 + x^{2} = - 2 \cdot 1 + 0 \cdot x + 1 \cdot x^{2} + 0 \cdot x^{3}$

Now,

$\begin{matrix} T (x^{2}) = x^{2} \cdot 2 - 2 \cdot 2 x + x \cdot x^{2} (Sin c e, p^{''} (x) = 2 a n d p^{'} (x) = 2 x) \\ = 2 x^{2} - 4 x + x^{3} \\ = - 4 x + 2 x^{2} + x^{3} \\ = 0 \cdot 1 - 4 \cdot x + 2 \cdot x^{2} + 1 \cdot x^{3} \end{matrix}$