Math Problem Statement
Let T : P2(R) → R2 be a linear transformation with bases α = {p1(x), p2(x), p3(x)} and β = {(2, 3), (−1, 5)} for P2(R) and R2 respectively. Suppose [T ]β α = [ 1 4 3 −1 1 5 ] (a) Find T (p3(x)). (b) Suppose p(x) = 3p1(x) + p2(x) − p3(x). Find T (p(x))
Solution
Let's tackle this problem step by step.
Part (a) - Finding :
We are given that the matrix representation of the linear transformation , denoted as , is:
\begin{bmatrix} 1 & 4 & 3 \\ -1 & 1 & 5 \end{bmatrix}$$ Here, $$\alpha = \{p_1(x), p_2(x), p_3(x)\}$$ is the basis for $$P_2(\mathbb{R})$$, and $$\beta = \{(2, 3), (-1, 5)\}$$ is the basis for $$\mathbb{R}^2$$. We need to find $$T(p_3(x))$$, which is the image of $$p_3(x)$$ under the transformation $$T$$. To do so, we use the fact that the columns of the matrix $$[T]_{\beta\alpha}$$ represent the coordinates of $$T(p_1(x))$$, $$T(p_2(x))$$, and $$T(p_3(x))$$ with respect to the basis $$\beta$$. The third column of $$[T]_{\beta\alpha}$$ corresponds to $$T(p_3(x))$$ in the basis $$\beta$$, and we have: $$[T(p_3(x))]_{\beta} = \begin{bmatrix} 3 \\ 5 \end{bmatrix}$$ This means that: $$T(p_3(x)) = 3(2, 3) + 5(-1, 5)$$ Now, let's compute this: $$T(p_3(x)) = 3(2, 3) + 5(-1, 5) = (6, 9) + (-5, 25) = (6 - 5, 9 + 25) = (1, 34)$$ Thus, the image of $$p_3(x)$$ under $$T$$ is: $$T(p_3(x)) = (1, 34)$$ ### Part (b) - Finding $$T(p(x))$$: Next, we are given that $$p(x) = 3p_1(x) + p_2(x) - p_3(x)$$. We need to find $$T(p(x))$$. By linearity of the transformation $$T$$, we have: $$T(p(x)) = T(3p_1(x) + p_2(x) - p_3(x)) = 3T(p_1(x)) + T(p_2(x)) - T(p_3(x))$$ We know the matrix representation $$[T]_{\beta\alpha}$$, which gives us the images of $$p_1(x)$$, $$p_2(x)$$, and $$p_3(x)$$ in the basis $$\beta$$: \[ [T(p_1(x))]_{\beta} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \quad [T(p_2(x))]_{\beta} = \begin{bmatrix} 4 \\ 1 \end{bmatrix}, \quad [T(p_3(x))]_{\beta} = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \] Using linearity, we calculate $$T(p(x))$$ as follows: $$T(p(x)) = 3\begin{bmatrix} 1 \\ -1 \end{bmatrix} + \begin{bmatrix} 4 \\ 1 \end{bmatrix} - \begin{bmatrix} 3 \\ 5 \end{bmatrix}$$ Now perform the arithmetic: $$T(p(x)) = \begin{bmatrix} 3 \\ -3 \end{bmatrix} + \begin{bmatrix} 4 \\ 1 \end{bmatrix} - \begin{bmatrix} 3 \\ 5 \end{bmatrix} = \begin{bmatrix} 3 + 4 - 3 \\ -3 + 1 - 5 \end{bmatrix} = \begin{bmatrix} 4 \\ -7 \end{bmatrix}$$ Thus, the image of $$p(x)$$ under $$T$$ is: $$T(p(x)) = (4, -7)$$ --- ### Summary of Answers: - (a) $$T(p_3(x)) = (1, 34)$$ - (b) $$T(p(x)) = (4, -7)$$ --- Would you like a detailed breakdown of any specific part? Feel free to ask! Here are some related questions for further exploration: 1. How does the matrix representation of a linear transformation relate to the change of basis? 2. What is the geometric interpretation of the linear transformation $$T$$ based on its matrix? 3. How do you compute the matrix of a linear transformation with respect to different bases? 4. How can you check if two linear transformations are equivalent? 5. What does it mean for a transformation to be diagonalizable, and how would you find the diagonalization of a matrix? **Tip**: When dealing with linear transformations and their matrix representations, always remember that the columns of the transformation matrix represent the transformed coordinates of the basis vectors in the target space.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Transformations
Matrix Representation
Vector Spaces
Change of Basis
Formulas
T(p(x)) = T(3p1(x) + p2(x) − p3(x))
Matrix multiplication for linear transformations
Theorems
Linearity of transformations
Matrix representation of linear transformations
Suitable Grade Level
Grades 11-12