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

step 1 First, we are going to evaluate T of the vector 2 -30. The first entry is given by the second co

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

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

Key Concepts

Linear Transformation

A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. It is often defined by a formula or matrix, and it transforms input vectors into output vectors while maintaining the structure of the space. This concept is central in linear algebra as it provides a framework to study and solve various problems involving changes of coordinates, maps, or relations between different vector spaces.

Image of a Vector under a Linear Transformation

The image of a vector under a linear transformation is obtained by applying the transformation rule to the vector. This means replacing the vector components into the transformation’s formula, resulting in a new vector. Computing the image is a fundamental step in understanding how the transformation acts on the space and in verifying properties like linearity or obtaining the range of the transformation.

Preimage (Inverse Image) under a Linear Transformation

The preimage, or inverse image, of a vector is the set of all vectors in the domain that are mapped to that specific vector in the codomain by the transformation. Finding the preimage typically involves solving a system of linear equations derived from the transformation's definition. This concept is crucial for identifying whether the transformation is one-to-one (injective) and for understanding the structure of the solution space associated with the transformation.

Systems of Linear Equations

Systems of linear equations arise when solving for the preimage of a vector under a linear transformation. Such systems consist of multiple equations that correspond to each component of the transformed vector. Solving these systems helps determine if there exists a unique solution, no solution, or infinitely many solutions, thus revealing important properties of the transformation like injectivity or the existence of a kernel.

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