Eigenvalues of linear transformations involving polynomials.Linear algebra **Matrix Representation of a Linear Transformation**This exercise involves completing the matrix representation for the linear transformation $ T : P_2 \rightarrow P_2 $, which acts on polynomials of degree 2 or less.### Given TransformationThe linear transformation $ T $ is defined as:\[T(ax^2 + bx + c) = (-3a + 7b + 3c)x^2 + (2a + 7b + 5c)x + (4a - 2b + 2c)\]### Matrix RepresentationTo find the matrix form of $ T $, express the transformation as a matrix multiplication:\[T \left( \begin{bmatrix} a \\ b \\ c \end{bmatrix} \right) = \begin{bmatrix}\text{[Ex: 5]} & & \\& & \\& & \end{bmatrix}\begin{bmatrix}a \\ b \\ c\end{bmatrix}\]#### Explanation of Entries- Each input polynomial $ ax^2 + bx + c $ is mapped to a new polynomial by calculating the coefficients of $ x^2 $, $ x $, and the constant term.- The matrix to be filled corresponds to these transformations whereby the entries will reflect coefficients that multiply $ a $, $ b $, and $ c $ in each term of $ T $.This exercise requires completing the empty cells to form the correct transformation matrix that encodes the given polynomial transformation rules.

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Answered: Complete the matrix representation for…

Complete the matrix representation for the linear transformation T: P₂ → P₂ given by T(ax² +bx+c) = (−3a + 7b + 3c)x² + (2a + 7b+5c)x + (4a − 2b +2c). a (ED с = Ex: 5 ↑ ŵ ŵ ♥ a b C

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

Eigenvalues of linear transformations involving polynomials.

Linear algebra

**Matrix Representation of a Linear Transformation**

This exercise involves completing the matrix representation for the linear transformation $ T : P_2 \rightarrow P_2 $, which acts on polynomials of degree 2 or less.

### Given Transformation

The linear transformation $ T $ is defined as:

\[
T(ax^2 + bx + c) = (-3a + 7b + 3c)x^2 + (2a + 7b + 5c)x + (4a - 2b + 2c)
\]

### Matrix Representation

To find the matrix form of $ T $, express the transformation as a matrix multiplication:

\[
T \left( \begin{bmatrix} a \\ b \\ c \end{bmatrix} \right) = \begin{bmatrix}
\text{[Ex: 5]} & & \\
& & \\
& &
\end{bmatrix}
\begin{bmatrix}
a \\ b \\ c
\end{bmatrix}
\]

#### Explanation of Entries

- Each input polynomial $ ax^2 + bx + c $ is mapped to a new polynomial by calculating the coefficients of $ x^2 $, $ x $, and the constant term.
- The matrix to be filled corresponds to these transformations whereby the entries will reflect coefficients that multiply $ a $, $ b $, and $ c $ in each term of $ T $.

This exercise requires completing the empty cells to form the correct transformation matrix that encodes the given polynomial transformation rules.

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.

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