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 want to evaluate t of 04, and this transformation is given by the following entries. T

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

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Key Concepts

Linear Transformation

A linear transformation is a mapping between vector spaces that preserves vector addition and scalar multiplication. It takes any vector from one space and transforms it into a vector in another space according to a defined rule, ensuring that the structure of the space is maintained.

Zero Vector

The zero vector serves as the additive identity in any vector space and has the special property that every linear transformation maps it to the zero vector of the target space. This is a consequence of the linearity property T(0) = 0.

Image of a Vector

The image of a vector under a transformation is the result of applying the transformation's rule to that vector. This concept is central in determining how vectors are repositioned in the target space, as it provides direct insight into the behavior of the transformation.

Kernel and Preimage

The kernel of a linear transformation consists of all vectors in the domain that map to the zero vector in the codomain. When finding the preimage of a given vector, especially the zero vector, the problem reduces to determining the set of vectors that the transformation sends to that vector, which is essential for understanding the structure and invertibility of the transformation.

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