I'm looking at page 292 of the Linear Algebra and Its Applications 5th Edition Textbook by David C. Lay. Could someone break down Example 2 some more for me? I don't understand how T(1) = 0, T(t) = 1, and T(t^2) = 2t or which function we should look t for the mapping. The entire part a section of that Example 2 problem I need help with. **Example 2**The mapping \( T: P_2 \to P_2 \) defined by \[ T(a_0 + a_1t + a_2t^2) = a_1 + 2a_2t \]is a linear transformation. (Calculus students will recognize \( T \) as the differentiation operator.)**a.** Find the \( B \)-matrix for \( T \), when \( B \) is the basis \(\{1, t, t^2\}\).**b.** Verify that \([T(p)]_B = [T]_B [p]_B \) for each \( p \) in \( P_2 \).**Solution****a.** Compute the images of the basis vectors:- \( T(1) = 0 \)- \( T(t) = 1 \)- \( T(t^2) = 2t \)Then write the \( B \)-coordinate vectors of \( T(1), T(t), \) and \( T(t^2) \) (which are found by inspection in this example) and place them together as the \( B \)-matrix for \( T \):\[[T(1)]_B = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \quad [T(t)]_B = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad [T(t^2)]_B = \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}\]The zero polynomial: A polynomial whose value is always 0.The polynomial whose value is always 1.

Let us consider the linear transformation T:V→W and B be the basis of the the vector space V Thus, the image of the basis is {T(v)} where v is in basis of the W The coefficient matrix is obtained by using the basis of W The image of every vector of V is written as the basis of W . The coefficient of the basis form the coefficient matrix Let us consider the given linear transformation T: P2→P2 defined as T(a0+a1x+a2x2)=a1+2a2x Here T is a differential operator a) Let B=1,t,t2 be the basis The image of the Basis is obtained as T(1)=T(1+0t+0t2)=0+2(0)t uisng defination=0T(t)=T(0+1t+0t2)=1+2(0)t uisng defination=1T(t2)=T(0+0t+1t2)=0+2(1)t uisng defination=2t b) Now, the coefficient matrix of the image of basis is T(1)=0+0t+0t2[T(1)]=000T(t)=1+0t+0t2[T(t)]=100T(t2)=0+2t+0t2[T(1)]=020 The coefficient matrix is [T]B=[T(1) T(t) T(t2)]=010002000

Answered: **Example 2** The mapping \( T: P_2…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

I'm looking at page 292 of the Linear Algebra and Its Applications 5th Edition Textbook by David C. Lay. Could someone break down Example 2 some more for me? I don't understand how T(1) = 0, T(t) = 1, and T(t^2) = 2t or which function we should look t for the mapping. The entire part a section of that Example 2 problem I need help with.

**Example 2**

The mapping \( T: P_2 \to P_2 \) defined by

\[ T(a_0 + a_1t + a_2t^2) = a_1 + 2a_2t \]

is a linear transformation. (Calculus students will recognize \( T \) as the differentiation operator.)

**a.** Find the \( B \)-matrix for \( T \), when \( B \) is the basis \(\{1, t, t^2\}\).

**b.** Verify that \([T(p)]_B = [T]_B [p]_B \) for each \( p \) in \( P_2 \).

**Solution**

**a.** Compute the images of the basis vectors:

- \( T(1) = 0 \)
- \( T(t) = 1 \)
- \( T(t^2) = 2t \)

Then write the \( B \)-coordinate vectors of \( T(1), T(t), \) and \( T(t^2) \) (which are found by inspection in this example) and place them together as the \( B \)-matrix for \( T \):

\[
[T(1)]_B = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \quad [T(t)]_B = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad [T(t^2)]_B = \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}
\]

The zero polynomial: A polynomial whose value is always 0.

The polynomial whose value is always 1.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

Expert Solution