What Makes a Vector Space? Exploring Subspaces, Dimensions, and Solution Sets

Vector Space

Let V be a non‑empty set of n‑dimensional vectors. If V is closed under vector addition and scalar multiplication—meaning that for any u, v in V, u+v is also in V, and for any scalar c and any v in V, c·v is in V—then V is called a vector space.

Subspace

Given a vector space V and a subset W, if W itself is closed under addition and scalar multiplication, then W is a subspace of V. For example, any set of vectors formed by selecting a subset of the coordinates of an n‑dimensional space forms a subspace of that space.

The zero subspace, consisting only of the zero vector, is always a non‑empty subspace of any vector space because it is trivially closed under the required operations.

Dimension

If a vector space V contains a set of k vectors that are linearly independent and every vector in V can be expressed as a linear combination of these k vectors, then the set forms a basis of V. The number k is called the dimension of V, and the space spanned by these basis vectors is the entire vector space.

Solution Vectors

A homogeneous linear system can be written in vector form as A·x = 0. Any vector x that satisfies this equation is called a solution vector of the system, and the collection of all such vectors constitutes the solution space of the homogeneous system.

Properties of Solution Vectors

(1) If x₁ and x₂ are solutions, then any linear combination α·x₁ + β·x₂ (with scalars α, β) is also a solution. (2) If x is a solution and c is any real number, then c·x is also a solution. These properties show that the set of all solution vectors is closed under vector addition and scalar multiplication, making it a vector space called the solution space of the homogeneous system.

Fundamental Solution Set

The set of all solutions to a homogeneous linear system forms a vector space. When the rank of the coefficient matrix is r, the dimension of the solution space is n − r, where n is the number of unknowns. A basis of this solution space is referred to as a fundamental solution set.