Let T: R³ → R² be defined by T -~--~--E) ----B]}~~(~-| ||-~ ~[]) Let B = u₁ U3 = { = [3], , = 6 (ED) ] -2x1 2x2 + x3. = Ex: 5 3 and C= What augmented matrix should be used to find [T], the matrix representation of Twith respect to the bases B and C.
Let T: R³ → R² be defined by T -~--~--E) ----B]}~~(~-| ||-~ ~[]) Let B = u₁ U3 = { = [3], , = 6 (ED) ] -2x1 2x2 + x3. = Ex: 5 3 and C= What augmented matrix should be used to find [T], the matrix representation of Twith respect to the bases B and 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
![Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) be defined by
\[
T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{bmatrix} -2x_1 \\ 2x_2 + x_3 \end{bmatrix}.
\]
Let \( \mathcal{B} = \left\{ \mathbf{u}_1 = \begin{bmatrix} -1 \\ -7 \\ 4 \end{bmatrix}, \mathbf{u}_2 = \begin{bmatrix} 3 \\ 1 \\ -5 \end{bmatrix}, \mathbf{u}_3 = \begin{bmatrix} -4 \\ 0 \\ -3 \end{bmatrix} \right\} \)
and
\(\mathcal{C} = \left\{ \mathbf{v}_1 = \begin{bmatrix} -7 \\ 6 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}. \)
What augmented matrix should be used to find \([ T ]_{\mathcal{C}}^{\mathcal{B}}\), the matrix representation of \( T \) with respect to the bases \(\mathcal{B}\) and \(\mathcal{C}\).
\[
\begin{bmatrix}
\text{Ex: 5} & & \\
& & \\
& &
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97fa71a9-ddb9-496b-9b0a-bf970e388fad%2Fb5b0c042-e271-4954-86d8-5c9ff5de68a0%2Frvjaty9_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) be defined by
\[
T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{bmatrix} -2x_1 \\ 2x_2 + x_3 \end{bmatrix}.
\]
Let \( \mathcal{B} = \left\{ \mathbf{u}_1 = \begin{bmatrix} -1 \\ -7 \\ 4 \end{bmatrix}, \mathbf{u}_2 = \begin{bmatrix} 3 \\ 1 \\ -5 \end{bmatrix}, \mathbf{u}_3 = \begin{bmatrix} -4 \\ 0 \\ -3 \end{bmatrix} \right\} \)
and
\(\mathcal{C} = \left\{ \mathbf{v}_1 = \begin{bmatrix} -7 \\ 6 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}. \)
What augmented matrix should be used to find \([ T ]_{\mathcal{C}}^{\mathcal{B}}\), the matrix representation of \( T \) with respect to the bases \(\mathcal{B}\) and \(\mathcal{C}\).
\[
\begin{bmatrix}
\text{Ex: 5} & & \\
& & \\
& &
\end{bmatrix}
\]
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