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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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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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 \( 4\mathbf{u} + 3\mathbf{v} \).

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The image of \( 4\mathbf{u} \) is \( \boxed{} \).

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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