Definition

1. Noun:
   • A space that is contained within another space: In mathematics, particularly linear algebra, a "subspace" is a set of vectors that forms a space itself but is entirely contained within a larger vector space. It must satisfy specific properties (like containing the zero vector and being closed under addition and scalar multiplication).

Usage Examples

• Noun:
  • The set of all points on a line through the origin is a subspace of the two-dimensional plane.
  • To solve the system, we first need to find a basis for the null subspace of the matrix.
  • The concept of a subspace is fundamental to understanding the structure of higher-dimensional spaces.

Advanced Usage

• Invariant Subspace: A subspace that is mapped to itself by a given linear transformation.
  • Finding the invariant subspaces of an operator helps in its decomposition.
• Proper Subspace: A subspace that is contained within but not equal to the larger space.
  • A line is a proper subspace of a plane.

Variants and Related Words

• Subspaces (n): The plural form of subspace.
  • The lecture covered the intersection and sum of subspaces.
• Subspace topology (n): In topology, the topology induced on a subset of a topological space.
  • The open sets in the subspace topology are defined relative to the larger space.

Synonyms

• Linear subspace: A more precise synonym, emphasizing the linear algebraic structure.
• Vector subspace: Another specific term used interchangeably with subspace in the context of vector spaces.

Related Phrases and Concepts

• To span a subspace: To generate a subspace from a set of vectors.
  • These two vectors span a two-dimensional subspace of R³.
• Dimension of a subspace: The number of vectors in a basis for the subspace.
  • The dimension of that particular subspace is three.

Noun

1. a space that is contained within another space