Finding the Image of a Subspace under a Linear Transformation

Consider the linear transformation that maps $\mathbb{R}^3$ to $\mathbb{R}^2$ given by $L(v) = (v_1, v_2 - v_3)$. Find the image of the subspace of $\mathbb{R}^3$ given by the vectors of length 3 where the second element is two times the first element, and the third element is 0.

In this problem, we explore the concept of linear transformations and their effects on subspaces. A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. When dealing with linear transformations, one crucial aspect to understand is how they map elements from one vector space to another, especially when applied to subspaces. In this problem, the transformation maps vectors from a three-dimensional space $\mathbb{R}^3$ to a two-dimensional space $\mathbb{R}^2$.

The given subspace consists of vectors in $\mathbb{R}^3$ that satisfy specific linear conditions. Understanding subspaces often requires familiarity with conditions that characterize a subset of a vector space and how these subsets remain invariant or change under linear maps. In this case, the subspace is described by vectors of a certain length that meet particular criteria related to their components.

The problem also introduces the concept of image, which, in the context of linear transformations, refers to the set of all possible output vectors when the transformation is applied to every vector in the domain. Finding the image of a subspace gives insights into what part of the target vector space is covered by transformed vectors. Thus, this exercise not only reinforces the understanding of linear maps but also sharpens skills in visualizing and calculating the effects of such maps on various subspaces.