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 $\overrightarrow{\mathbf{w}}$. $T\left(u_1, u_2, u_3\right)=\left(u_1, u_2, u_1+u_3\right)$, $$ \overline{\mathbf{u}}=(1,2,1), \quad \overrightarrow{\mathbf{w}}=(0,0,1) $$

Problems $4 I-48$. compute the image under $T$ of $\overrightarrow{\mathrm{u}}$, and find the vector(s), if any, that are mapped to $\overrightarrow{\mathbf{w}}$.
$T\left(u_1, u_2, u_3\right)=\left(u_1, u_2, u_1+u_3\right)$,
$$
\overline{\mathbf{u}}=(1,2,1), \quad \overrightarrow{\mathbf{w}}=(0,0,1)
$$

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Key Concepts

Linear Transformation

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. It is typically represented by a formula or a matrix and maps vectors from one space to another without altering the inherent linear structure of the space. In the given context, the transformation takes a vector from R^3 and produces another vector in R^3 following specific coordinate rules.

Image of a Vector

The image of a vector under a linear transformation is the output obtained when the transformation is applied to that specific vector. This involves substituting the components of the given vector into the transformation’s formula to compute the resulting vector in the codomain.

Pre-image or Inverse Image

In the context of a linear transformation, the pre-image (or inverse image) of a vector is the set of all vectors in the domain that are mapped to a particular vector in the codomain. Finding the pre-image often entails solving a system of linear equations derived from equating the transformation’s output to the target vector. This step is crucial when determining whether there are vectors that map to a given vector, and if so, the complete description of that set.

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