Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

Help me solve this linear algebra problem to review for my test.

Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation that maps \( \mathbf{u} = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \) into \( \begin{bmatrix} 6 \\ 1 \end{bmatrix} \) and maps \( \mathbf{v} = \begin{bmatrix} 4 \\ 4 \end{bmatrix} \) into \( \begin{bmatrix} -1 \\ 4 \end{bmatrix} \). Use the fact that \( T \) is linear to find the images under \( T \) of \( 4\mathbf{u}, 3\mathbf{v}, \) and \( 4\mathbf{u} + 3\mathbf{v} \). --- The image of \( 4\mathbf{u} \) is \( \boxed{} \).
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Transcribed Image Text:Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation that maps \( \mathbf{u} = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \) into \( \begin{bmatrix} 6 \\ 1 \end{bmatrix} \) and maps \( \mathbf{v} = \begin{bmatrix} 4 \\ 4 \end{bmatrix} \) into \( \begin{bmatrix} -1 \\ 4 \end{bmatrix} \). Use the fact that \( T \) is linear to find the images under \( T \) of \( 4\mathbf{u}, 3\mathbf{v}, \) and \( 4\mathbf{u} + 3\mathbf{v} \). --- The image of \( 4\mathbf{u} \) is \( \boxed{} \).
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