Let T: R² R² be defined by T Let B := { m₁ = [ ^ ^"] , ' = [ 5 ] } am³ € = {v₁ = [1] · v₂ = [7²]} and V2 ¹([2]) = [²₂] x1 X2 Find [T], the matrix representation of Twith respect to the bases B and C. [T = Ex: 5 +
Let T: R² R² be defined by T Let B := { m₁ = [ ^ ^"] , ' = [ 5 ] } am³ € = {v₁ = [1] · v₂ = [7²]} and V2 ¹([2]) = [²₂] x1 X2 Find [T], the matrix representation of Twith respect to the bases B and C. [T = Ex: 5 +
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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Hi can I please get some help with this problem, the steps are unclear to me. Thanks!
![On this educational website, we explore the transformation \( T \) defined over \(\mathbb{R}^2 \to \mathbb{R}^2 \). The transformation \( T \) is given by:
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} x_1 \\ x_1 - x_2 \end{bmatrix}.
\]
We are provided with two bases:
For basis \( \mathcal{B} \):
\[
\mathcal{B} = \left\{ \mathbf{u}_1 = \begin{bmatrix} -4 \\ 3 \end{bmatrix}, \mathbf{u}_2 = \begin{bmatrix} 5 \\ -1 \end{bmatrix} \right\}
\]
For basis \( \mathcal{C} \):
\[
\mathcal{C} = \left\{ \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} -2 \\ 1 \end{bmatrix} \right\}
\]
The goal is to find \([T]_{\mathcal{B}}^{\mathcal{C}}\), which is the matrix representation of \( T \) with respect to the bases \( \mathcal{B} \) and \( \mathcal{C} \).
The resulting matrix is to be entered into an input cell, indicating the various entries for calculation.
For clarity, there's a 2x2 matrix template shown, with the first entry placeholder highlighted and labeled as "Ex: 5".
The matrix representation to be found is denoted by:
\[
[T]_{\mathcal{B}}^{\mathcal{C}} = \begin{bmatrix} \boxed{\phantom{00}} & \boxed{\phantom{00}} \\ \boxed{\phantom{00}} & \boxed{\phantom{00}} \end{bmatrix}
\]
This concise, conceptual matrix indicates the transformational relationships between vectors in these bases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8aa82a-31a1-4382-a30e-3ad6a3174f84%2F5052e46e-d828-42fb-a4b0-caa712e23379%2Fzoph755_processed.png&w=3840&q=75)
Transcribed Image Text:On this educational website, we explore the transformation \( T \) defined over \(\mathbb{R}^2 \to \mathbb{R}^2 \). The transformation \( T \) is given by:
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} x_1 \\ x_1 - x_2 \end{bmatrix}.
\]
We are provided with two bases:
For basis \( \mathcal{B} \):
\[
\mathcal{B} = \left\{ \mathbf{u}_1 = \begin{bmatrix} -4 \\ 3 \end{bmatrix}, \mathbf{u}_2 = \begin{bmatrix} 5 \\ -1 \end{bmatrix} \right\}
\]
For basis \( \mathcal{C} \):
\[
\mathcal{C} = \left\{ \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} -2 \\ 1 \end{bmatrix} \right\}
\]
The goal is to find \([T]_{\mathcal{B}}^{\mathcal{C}}\), which is the matrix representation of \( T \) with respect to the bases \( \mathcal{B} \) and \( \mathcal{C} \).
The resulting matrix is to be entered into an input cell, indicating the various entries for calculation.
For clarity, there's a 2x2 matrix template shown, with the first entry placeholder highlighted and labeled as "Ex: 5".
The matrix representation to be found is denoted by:
\[
[T]_{\mathcal{B}}^{\mathcal{C}} = \begin{bmatrix} \boxed{\phantom{00}} & \boxed{\phantom{00}} \\ \boxed{\phantom{00}} & \boxed{\phantom{00}} \end{bmatrix}
\]
This concise, conceptual matrix indicates the transformational relationships between vectors in these bases.
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