Glossary

3.1 Vector Space Axioms and Subspaces

3 min read•july 30, 2024

Vector spaces are the backbone of linear algebra, providing a framework for understanding linear relationships. They're defined by specific rules called axioms, which govern how vectors interact through addition and multiplication.

Subspaces are subsets of vector spaces that follow the same rules. They're crucial for breaking down complex problems into simpler parts. Understanding subspaces helps you grasp the structure of vector spaces and solve linear equations more effectively.

Vector Spaces and Properties

Definition and Axioms

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  • comprises a set V of vectors with two operations
  • Satisfies specific axioms over a field (typically real numbers ℝ or complex numbers ℂ)
  • Ten vector space axioms ensure mathematical structure
    • and scalar multiplication
    • Commutativity and
    • : 1v=v1v = v
    • : 0v=00v = 0

Dimensions and Examples

  • denotes number of vectors in the
    • Basis consists of spanning the entire space
  • Common vector space examples
    • ℝⁿ (n-dimensional real space)
    • Function spaces (continuous functions on an interval)
    • Polynomial spaces (polynomials of degree ≤ n)
  • Abstract vector spaces extend beyond geometric vectors
    • Vectors can represent functions, matrices, or other mathematical objects

Verifying Vector Spaces

Axiom Verification Process

  • Check all ten vector space axioms for given set and operations
  • Test closure property for addition and scalar multiplication
    • Ensure operations result in elements within the set
  • Verify commutativity and associativity of vector addition
    • Use arbitrary elements from the set
  • Demonstrate existence of unique zero vector and additive inverses
  • Confirm distributive properties
    • Scalar multiplication over vector addition
    • Scalar multiplication over field addition
  • Pay special attention to scalar identity and zero scalar properties

Disproving Vector Spaces

  • Use counter-examples to disprove vector space status
  • Identify specific axiom violations
    • Example: Set of positive real numbers fails zero vector axiom
  • Analyze edge cases and boundary conditions
    • Example: Set of integers under real scalar multiplication not closed

Subspaces of Vector Spaces

Subspace Definition and Properties

  • consists of subset W of vector space V
    • Forms vector space under same operations as V
  • Three conditions for subspace verification
    • W is non-empty
    • W is closed under vector addition
    • W is closed under scalar multiplication
  • Zero vector of original space must be in subspace
    • Serves as additive identity for subspace
  • Common subspace types
    • Null spaces (solutions to Ax = 0)
    • Column spaces (span of matrix columns)
    • Row spaces (span of matrix rows)

Subspace Relationships

  • Intersection of two subspaces always forms subspace
  • generally not a subspace
    • Exception: One subspace contained within the other
  • Span of vector set from original space always creates subspace
  • combines three conditions into single statement
    • ∀u,v∈W,∀c∈F,cu+v∈W\forall u, v \in W, \forall c \in \mathbb{F}, cu + v \in W
    • F represents the underlying field (ℝ or ℂ)

Subspace Identification

Verification Process

  • Check three subspace conditions
    • Non-emptiness
    • Closure under addition
  • Verify in subset
    • Proves non-emptiness
    • Partially addresses closure
  • Test addition closure
    • Show sum of arbitrary elements remains in subset
  • Demonstrate scalar multiplication closure
    • Prove scalar multiple of any element stays in subset

Special Cases and Considerations

  • Analyze sets defined by equations or conditions
    • Show conditions preserved under vector space operations
  • Use counter-examples to disprove subspace status efficiently
    • Example: Plane not passing through origin fails zero vector condition
  • Examine subsets defined by strict inequalities
    • Often fail subspace criteria due to zero vector exclusion
    • Example: x∈R3:x1+x2+x3>0{x \in \mathbb{R}^3 : x_1 + x_2 + x_3 > 0} not a subspace
  • Consider geometric interpretations
    • Subspaces as lines, planes, or hyperplanes through origin

Key Terms to Review (26)

Associativity of Addition: The associativity of addition states that the way in which numbers are grouped when being added does not change the sum. This property ensures that for any three elements in a vector space, say \(u\), \(v\), and \(w\), the equation \(u + (v + w) = (u + v) + w\) holds true. This characteristic plays a critical role in ensuring consistency in calculations and operations within vector spaces.
Basis: A basis is a set of vectors in a vector space that are linearly independent and span the entire space. This means that every vector in the space can be expressed as a unique linear combination of the basis vectors, providing a way to represent and analyze the structure of the vector space.
Closure Under Addition: Closure under addition means that if you take any two elements from a set, their sum will also be an element of that same set. This property is crucial for determining whether a set is a subspace of a vector space, as it ensures that the addition of vectors within the set doesn't lead to an element outside of it. In the context of vector spaces, closure under addition supports the structure necessary for forming linear combinations and establishes foundational relationships among vectors.
Closure under Scalar Multiplication: Closure under scalar multiplication refers to the property that if a vector is in a set and it is multiplied by a scalar, the resulting vector also belongs to the same set. This concept is fundamental in understanding vector spaces, as it helps establish whether a collection of vectors can be classified as a subspace. If a set is closed under scalar multiplication, it ensures that scaling vectors maintains the integrity of the vector space's structure.
Column Space: The column space of a matrix is the set of all possible linear combinations of its column vectors. This space represents all the vectors that can be formed by combining the columns of the matrix, making it crucial in understanding the solutions of linear equations and the effects of linear transformations.
Commutativity of Addition: Commutativity of addition refers to the property that states the order in which two elements are added does not affect the sum. This fundamental concept is essential in vector spaces, as it helps to define how vectors can be combined. Understanding this property is crucial because it allows for flexibility in operations involving vectors, ensuring that regardless of their arrangement, the outcome remains consistent and predictable.
Dimension: Dimension is a fundamental concept that represents the number of coordinates needed to specify a point within a space. It helps us understand the size and structure of vector spaces and subspaces, influencing how we visualize and manipulate mathematical objects. A space with a higher dimension often contains more complex relationships and transformations than a lower-dimensional space.
Distributivity of Scalar Multiplication over Field Addition: Distributivity of scalar multiplication over field addition states that for any scalar values and vectors in a vector space, the operation distributes over addition. In simpler terms, if you have a scalar multiplying a sum of two vectors, it is the same as multiplying each vector by the scalar and then adding the results together. This property is crucial in defining how operations work in a vector space, ensuring consistency and coherence in mathematical manipulations.
Distributivity of Scalar Multiplication over Vector Addition: The distributivity of scalar multiplication over vector addition states that for any scalar 'c' and any vectors 'u' and 'v', the equation $c(u + v) = cu + cv$ holds true. This property illustrates how scaling a sum of vectors is equivalent to scaling each vector individually and then adding the results together, which emphasizes the interaction between scalar multiplication and vector addition within the structure of a vector space.
Existence of Additive Inverses: The existence of additive inverses refers to the property that for every element in a vector space, there is another element (the additive inverse) such that their sum equals the zero vector. This concept is crucial in understanding how vector spaces function, as it ensures that every element can be 'canceled out' by its corresponding inverse, allowing for the completion of operations within the space.
Existence of Zero Vector: The existence of a zero vector in a vector space refers to the requirement that there is a unique vector, denoted as 0, which acts as the additive identity. This means that for any vector v in the space, when you add the zero vector to v, the result is v itself. The zero vector is crucial because it ensures that vector addition has an identity element, which is one of the fundamental properties required for a set to qualify as a vector space.
Intersection of Subspaces: The intersection of subspaces refers to the set of all vectors that are common to two or more subspaces within a vector space. This concept is vital because it helps in understanding how different subspaces relate to one another, particularly in terms of their shared elements and dimensionality. Analyzing intersections can reveal important properties about the overall structure of the vector space and can be crucial in applications involving linear transformations and systems of equations.
Linearly Independent Vectors: Linearly independent vectors are a set of vectors in a vector space where no vector can be expressed as a linear combination of the others. This means that if the only way to write a linear combination of these vectors equal to the zero vector is by having all coefficients equal to zero, they are considered linearly independent. The concept is crucial in understanding the structure of vector spaces and helps determine the dimension of a subspace.
Non-empty subset: A non-empty subset is a collection of elements derived from a larger set that contains at least one element. This term is crucial when discussing vector spaces because it helps to identify subsets that may retain certain properties of the larger set, such as closure under addition and scalar multiplication, which are essential for forming subspaces.
Null Space: The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector. This concept is important because it helps in understanding solutions to linear systems, particularly those that are homogeneous. The null space reveals the relationships among the columns of the matrix and is crucial for identifying linear dependencies and the dimension of subspaces.
Row Space: The row space of a matrix is the set of all linear combinations of its row vectors. It forms a subspace of the vector space, capturing the dimensions and directions that the rows of the matrix can span. The row space is essential for understanding the solutions to systems of linear equations and the relationship between a matrix and its rank.
Scalar Identity Property: The scalar identity property states that for any vector in a vector space, multiplying it by the scalar one leaves the vector unchanged. This property highlights the significance of the scalar multiplication operation within a vector space, as it reinforces the idea that the vector remains in the same position and retains its magnitude and direction when scaled by one. This concept is essential for understanding how vectors operate under scalar multiplication and forms part of the foundational axioms that define a vector space.
Scalar Multiplication: Scalar multiplication is an operation that involves multiplying a vector by a scalar (a single number), resulting in a new vector that points in the same or opposite direction depending on the sign of the scalar. This operation is fundamental in linear algebra as it helps to stretch or shrink vectors and changes their magnitude without altering their direction if the scalar is positive. Additionally, it plays a crucial role in understanding matrix operations and vector spaces.
Span of a set: The span of a set of vectors is the collection of all possible linear combinations of those vectors. This means it includes every vector that can be formed by taking any scalar multiple of the vectors in the set and adding them together. Understanding the span is crucial because it helps define the extent and limitations of the space that those vectors can cover, which directly relates to concepts like vector spaces and subspaces.
Subspace: A subspace is a subset of a vector space that is also a vector space itself, meaning it satisfies the same axioms of addition and scalar multiplication as the larger space. Subspaces must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. Understanding subspaces is crucial for grasping how linear transformations behave, especially in terms of their kernel and range, as these are specific types of subspaces that arise from applying linear transformations to vectors.
Subspace Test: The subspace test is a method used to determine if a subset of a vector space is itself a subspace. This involves checking three specific criteria: the zero vector must be in the subset, the subset must be closed under vector addition, and it must also be closed under scalar multiplication. If all three conditions are met, the subset qualifies as a subspace, which is essential for understanding the structure of vector spaces.
Union of Subspaces: The union of subspaces is the set containing all elements that belong to either one subspace or another, or both. However, it is crucial to understand that while individual subspaces can be combined, their union is not necessarily a subspace itself unless one is contained within the other. This concept is essential in understanding the behavior and relationships of vector spaces and their properties.
Vector Addition: Vector addition is the process of combining two or more vectors to produce a resultant vector. This operation adheres to specific rules and properties that make it a fundamental part of vector spaces, including commutativity and associativity, which facilitate the study of subspaces. Understanding vector addition is crucial for analyzing geometrical and physical situations in mathematics and physics, where direction and magnitude are important.
Vector Space: A vector space is a collection of vectors that can be added together and multiplied by scalars, satisfying certain axioms such as closure, associativity, and distributivity. This structure allows for linear combinations of vectors, which are crucial for understanding concepts like subspaces, transformations, and linear independence. The properties of vector spaces enable the exploration of linear equations, geometric interpretations, and the analysis of transformations in mathematics.
Zero Scalar Property: The zero scalar property states that when any vector in a vector space is multiplied by the scalar zero, the result is the zero vector of that vector space. This property is crucial as it reinforces the concept that the zero vector acts as the additive identity and plays an important role in the structure of vector spaces, ensuring consistency in operations involving scalars and vectors.
Zero vector presence: Zero vector presence refers to the inclusion of the zero vector in a vector space, which is defined as the unique vector that has a magnitude of zero and serves as the additive identity. This means that when the zero vector is added to any vector in the space, the result is that same vector, preserving the structure of the vector space. Its existence is crucial because it satisfies one of the fundamental axioms of vector spaces and underpins many properties of linear combinations and subspaces.
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