Existence of a Two-Dimensional Vector Space with Exactly Two Vectors

The concept of vector spaces is fundamental in linear algebra. A vector space over a field has a minimum set of elements that includes the zero vector and potentially other vectors. Surprisingly, there exists a vector space with only two vectors. This space is unique and interesting due to its simplicity and the constraints it imposes. This article will explore the existence and construction of a two-dimensional vector space with exactly two vectors.

Trivial Vector Space

The simplest example of a vector space with only two vectors is the trivial vector space, which is the space containing only the zero vector, denoted as {0}. This space is considered trivial because it doesn't provide much in the way of structure or operations for vectors beyond the zero vector. It serves as a foundational example but isn't generally of practical interest.

Vector Spaces Over Finite Fields

To find a more interesting example, we can look at vector spaces over finite fields. Specifically, let's consider the finite field GF(2), which is the field with only two elements, namely 0 and 1. Over this finite field, we can construct a vector space with exactly two vectors.

When we consider a vector space over GF(2) of dimension 1, it contains the zero vector (0) and one additional vector that can be any non-zero vector (1 in this case). Consequently, the vector space has exactly two vectors: the zero vector and the non-zero vector. We can denote this vector space as {0, 1}.

Unique Construction and Isomorphism

James Buddenhagen provided an excellent example of a vector space over GF(2). A vector space over a field F must include scalar multiplication by elements of F. Therefore, if the field has more than two elements, the vector space will have more than two vectors. This leaves us with the field GF(2), which has exactly two elements.

Let's consider a nonzero vector space V over GF(2). If we take any nonzero vector v ∈ V, then any scalar multiple of v (other than 0 and v) will also be in V. However, this leads to the conclusion that V must have at least three vectors, which contradicts our requirement that V has only two vectors.

Therefore, the only field that allows for such a vector space is GF(2). In this case, V is isomorphic to GF(2) as a linear space over itself. We can take the set {v} as a basis for V, where v ≠ 0. This confirms that V is a one-dimensional vector space over GF(2).

Conclusion

In summary, while vector spaces must contain the zero vector and can be constructed over finite fields, the only vector space that has exactly two vectors is the trivial vector space with the zero vector and a single non-zero vector from GF(2). This special vector space is isomorphic to GF(2) and provides a unique example of a vector space with minimal elements.

Understanding the existence and construction of such a vector space offers insights into the fundamental properties of vector spaces and the constraints that finite fields can impose on these structures.