Understanding Vector Spaces and Subspaces: Clarifying R2 in R3

When discussing vector spaces and their subspaces, especially in the context of (mathbb{R}^2) and (mathbb{R}^3), it is crucial to clarify some common misconceptions and subtle nuances. In this article, we will delve into the relationship between these spaces, addressing the question of whether (mathbb{R}^2) is a subset of (mathbb{R}^3) and exploring the concept of isomorphism.

Basic Definitions and Clarifications

1. Vector Spaces: Before diving into the specifics, it is important to establish the foundational definitions. A vector space over a field F (usually the real numbers (mathbb{R})) comprises a set of elements together with two operations: vector addition and scalar multiplication that satisfy certain axioms.

2. Subspaces: A subspace of a vector space is itself a vector space that is a subset of the original vector space. It must also satisfy the vector space axioms under the same operations as the parent space.

R2 and R3: Not Subspaces in a Natural Sense

One of the primary points of confusion is the relationship between (mathbb{R}^2) and (mathbb{R}^3). Specifically, (mathbb{R}^2) is not a subset of (mathbb{R}^3), and thus, it is not a subspace of (mathbb{R}^3). If we consider (mathbb{R}^2) as the set of all ordered pairs of real numbers, and (mathbb{R}^3) as the set of all ordered triples, these sets are fundamentally different because (mathbb{R}^2) has dimension 2, while (mathbb{R}^3) has dimension 3.

However, (mathbb{R}^2) can be isomorphic to infinitely many subspaces of (mathbb{R}^3). For instance, the set {xy0 | xy in R} is a subspace of (mathbb{R}^3) which is isomorphic to (mathbb{R}^2). This set is often referred to as the xy-plane in (mathbb{R}^3), and it can be mapped to (mathbb{R}^2) in a way that preserves the vector space structure.

Over in the Context of Vector Spaces

The term "over" in mathematics often refers to the field of scalars used in scalar multiplication. When we speak of (mathbb{R}^2) or (mathbb{R}^3) as vector spaces, we are implicitly assuming that the scalars are from the real numbers (mathbb{R}). This is what we mean when we say that (mathbb{R}^2) or (mathbb{R}^3) are vector spaces over the real numbers.

Embedding R2 in R3

While (mathbb{R}^2) can be embedded in (mathbb{R}^3), the embedding is not "natural" in the sense that the xy-plane {xy0 in (mathbb{R}^3)} is a natural subspace of (mathbb{R}^3). More formally, embedding (mathbb{R}^2) into (mathbb{R}^3) involves selecting two linearly independent vectors and considering their linear combinations, which can be done in infinitely many ways.

However, if you have a concrete setup with more structure, the embedding might appear natural within that context. For instance, you might define a particular way to map points from (mathbb{R}^2) to (mathbb{R}^3) based on specific conditions in your problem.

Conclusion

In summary, while it is possible to find isomorphic subspaces of (mathbb{R}^3) that are isomorphic to (mathbb{R}^2), (mathbb{R}^2) is not a subset of (mathbb{R}^3). The term "over" in this context refers to the field of scalars used in scalar multiplication, and when discussing vector spaces, we implicitly assume that the scalars are from the real numbers.

Understanding these nuances is essential for clear communication in mathematical discourse.