-1 --[*][*]and [:] [^^] into and maps v = into 5 4 Let T: R² R² be a linear transformation that maps u = . Use the fact that T is linear to find the images under T of 4u, 3v, and 4u + 3v.
-1 --[*][*]and [:] [^^] into and maps v = into 5 4 Let T: R² R² be a linear transformation that maps u = . Use the fact that T is linear to find the images under T of 4u, 3v, and 4u + 3v.
-1 --[*][*]and [:] [^^] into and maps v = into 5 4 Let T: R² R² be a linear transformation that maps u = . Use the fact that T is linear to find the images under T of 4u, 3v, and 4u + 3v.
Help me solve this linear algebra problem to review for my test.
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 \( 4\mathbf{u} + 3\mathbf{v} \).
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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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