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 $\overrightarrow{\mathbf{w}}$.
$T(x, y)=(x+y, x), \quad \overrightarrow{\mathbf{u}}=(1,0), \quad \overrightarrow{\mathbf{w}}=(3,1)$

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 can often be represented by a matrix and is characterized by its ability to map vectors from one space to another while maintaining the structure of the vector space.

Image of a Vector under a Linear Transformation

The image of a vector under a linear transformation is the result of applying the transformation to that vector. It represents where the original vector is sent in the codomain. Computing the image involves replacing the vector's components in the transformation’s formula or matrix representation.

Preimage (Inverse Image) of a Vector

The preimage of a vector under a linear transformation consists of all vectors in the domain that are mapped to a given vector in the codomain. Determining the preimage involves solving for the vectors that satisfy the equation defined by the transformation, which may involve solving a system of linear equations.

Solving Systems of Linear Equations

Solving systems of linear equations is a fundamental technique used to find unknown variables that satisfy multiple linear equations simultaneously. In the context of linear transformations, this method is applied to determine the vector(s) in the domain that map to a particular vector in the codomain.

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