Problems 4 I-48. compute the image under T of u, and find the vector(s), if any, that are mapped

step 1 In this problem we have to use the function to find the image of V and then the pre -image of W

Problems 4 I-48. compute the image under T of u, and find...

Problems $4 I-48$. compute the image under $T$ of $\overrightarrow{\mathrm{u}}$, and find the vector(s), if any, that are mapped to $\overrightarrow{\mathbf{w}}$. $T\left(u_1, u_2\right)=\left(u_1, u_1+2 u_2\right), \quad \overrightarrow{\mathbf{u}}=(1,2), \quad \overrightarrow{\mathbf{w}}=(1,3)$

Problems $4 I-48$. compute the image under $T$ of $\overrightarrow{\mathrm{u}}$, and find the vector(s), if any, that are mapped to $\overrightarrow{\mathbf{w}}$.
$T\left(u_1, u_2\right)=\left(u_1, u_1+2 u_2\right), \quad \overrightarrow{\mathbf{u}}=(1,2), \quad \overrightarrow{\mathbf{w}}=(1,3)$

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Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. This fundamental concept in linear algebra is used to study mappings between spaces, often represented by matrices, and to analyze their structural properties such as the rank, kernel, and image.

Image of a Vector under a Transformation

The image of a vector under a transformation is the output obtained when the linear transformation is applied to that vector. This concept helps in understanding how input vectors are manipulated and provides insight into the behavior of the transformation across the entire space.

Preimage (Inverse Image) and Solving Linear Equations

The preimage of a vector under a transformation is the set of all vectors that are mapped to that vector by the transformation. Determining this set often involves solving a system of linear equations, which is essential for understanding the conditions for the existence and uniqueness of solutions in the context of the transformation.

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