Subspace

Definition

A subspace is a subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication defined in the larger space. It must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. This concept plays a crucial role in understanding the structure and properties of normed linear spaces.

5 Must Know Facts For Your Next Test

1. Every vector space has at least two subspaces: the trivial subspace containing only the zero vector and the vector space itself.
2. A subspace must satisfy three criteria: it must contain the zero vector, it must be closed under addition, and it must be closed under scalar multiplication.
3. The intersection of any collection of subspaces is also a subspace.
4. If U and V are both subspaces of a vector space W, then the sum U + V (the set of all vectors that can be expressed as u + v where u ∈ U and v ∈ V) is also a subspace.
5. The dimension of a subspace can be less than or equal to the dimension of the original vector space.

Review Questions

• How do you determine if a subset is a subspace of a given vector space?
  • To determine if a subset is a subspace, you need to check three main conditions: first, ensure that the zero vector is included in the subset. Second, verify that for any two vectors in the subset, their sum is also in the subset, confirming closure under addition. Finally, check that multiplying any vector in the subset by a scalar results in another vector that is still within the subset, confirming closure under scalar multiplication. If all three conditions are met, then the subset is a subspace.
• Explain how the concept of subspaces relates to linear combinations and bases in a vector space.
  • Subspaces are directly linked to linear combinations because any vector in a subspace can be expressed as a linear combination of its basis vectors. A basis for a subspace consists of linearly independent vectors that span that particular subspace. Understanding how these vectors work together allows us to comprehend how larger spaces can be broken down into simpler parts through their subspaces. Essentially, knowing about bases helps in identifying and working with different subspaces within a given vector space.
• Evaluate the significance of subspaces in studying properties of normed linear spaces, including their impact on dimensions and convergence.
  • Subspaces are essential in understanding normed linear spaces because they allow us to analyze smaller, manageable parts of larger spaces while maintaining essential properties. The dimensionality of subspaces provides insight into how 'complex' or 'rich' a larger space can be, with certain dimensions indicating potential simplifications for calculations or proofs. Moreover, concepts like convergence can be examined within subspaces to determine whether sequences or series behave well—essentially assessing continuity and limits—without losing sight of broader implications in the entire normed linear space.

Related terms

Vector Space: A collection of vectors where vector addition and scalar multiplication are defined and satisfy specific axioms such as associativity, commutativity, and the existence of an additive identity.

Linear Combination: A sum of scalar multiples of vectors. A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the others.

Basis: A set of linearly independent vectors in a vector space that spans the entire space, meaning any vector in the space can be expressed as a linear combination of the basis vectors.