91Ó°ÊÓ

Linear transformations are fundamental operations in mathematics that map one vector space to another while preserving the structure of the space. Let's break it down in simpler terms. A linear transformation, say \( f: \mathbb{R}^n \to \mathbb{R}^m \), takes any vector \( \mathbf{x} \) in \( \mathbb{R}^n \) and maps it to a vector \( f(\mathbf{x}) \) in \( \mathbb{R}^m \).The key properties of a linear transformation include:

• Preservation of vector addition: \( f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) \).
• Preservation of scalar multiplication: \( f(c\mathbf{x}) = c f(\mathbf{x}) \) for any scalar \( c \).

These properties ensure that the "shape" or "linear nature" of the space is maintained, even though the actual dimensions or shapes might differ in the resulting space. In the given problem, applying \( f \) to an affine set \( S \) preserves its affine structure, which will be crucial to understand the transformation of subspaces and the nature of their translations.

An affine subspace is a set that resembles a linear subspace but is not necessarily passing through the origin. It can be viewed as a "shifted" version of a linear subspace. Formally, if we have a point \( \mathbf{p} \) and a linear subspace \( T \), then the affine subset \( S \) of \( \mathbb{R}^n \) can be described as \( S = \{ \mathbf{x} = \mathbf{p} + \mathbf{t} : \mathbf{t} \in T \} \).Key features of affine subspaces include:

• Every point \( \mathbf{x} \) in the affine subspace can be expressed as a sum of a fixed vector \( \mathbf{p} \) and a vector \( \mathbf{t} \) from a linear subspace \( T \).
• A translation by any vector in \( \mathbb{R}^n \) differs from moving along a linear subspace because it doesn't necessitate passing through the origin.

This concept is crucial when proving that \( f(S) \) is also affine, as it involves showing how a linear transformation retains the qualities of being an affine subspace, even in a different dimension.

Translating a subspace involves shifting it by adding a fixed vector, which allows the subspace to "move" around its environment. For example, suppose \( T \) is a linear subspace of \( \mathbb{R}^n \). A translation of \( T \) by a vector \( \mathbf{p} \) results in the affine subspace \( S = \mathbf{p} + T \).In the context of the exercise, when applying a linear transformation \( f \) to an affine subspace \( S = \mathbf{p} + T \), each element in \( S \) maps to the following in \( \mathbb{R}^m \):

• \( f(\mathbf{p} + \mathbf{t}) = f(\mathbf{p}) + f(\mathbf{t}) \).
• Here, \( f(\mathbf{p}) \) serves as the translation vector in \( \mathbb{R}^m \).
• The resulting subspace \( f(T) \) in \( \mathbb{R}^m \) retains linearity because of the properties of \( f \).

This means \( f(S) = f(\mathbf{p}) + f(T) \), demonstrating the translation of the subspace \( f(T) \) by \( f(\mathbf{p}) \), thus confirming that \( f(S) \) forms an affine subset in the new space.